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A Minchenko et al (2014) ldquoReductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized
Linear Differential Equations rdquo
International Mathematics Research Notices Vol 2014 Article ID rnt344 61 pages
doi101093imrnrnt344
Reductive Linear Differential Algebraic Groups and the Galois Groups of
Parameterized Linear Differential Equations
Andrey Minchenko1 Alexey Ovchinnikov23 and Michael F Singer4
1The Weizmann Institute of Science Department of Mathematics Rehovot 7610001 Israel2 Department of Mathematics CUNY Queens College 65-30 Kissena Blvd Queens NY
11367 USA 3 Department of Mathematics CUNY Graduate Center 365 Fifth Avenue New
York NY 10016 USA and 4 Department of Mathematics North Carolina State University
Raleigh NC 27695-8205 USA
Correspondence to be sent to aovchinnikovqccunyedu
We develop the representation theory for reductive linear differential algebraic groups (LDAGs) In particular we
exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs
extending existing results which were obtained for SL2 in the case of just one derivation As an application of the
above bound we develop an algorithm that tests whether the parameterized differential Galois group of a system
of linear differential equations is reductive and if it is calculates it
1 Introduction
At the most basic level a linear differential algebraic group (LDAG) is a group of matrices whose
entries are functions satisfying a fixed set of polynomial differential equations An algebraic study
of these objects in the context of differential algebra was initiated by Cassidy in [8] and further
developed by Cassidy [9 10 13 11 12] This theory of LDAGs has been extended to a theory
of general differential algebraic groups by Kolchin Buium Pillay and others Nonetheless inter-
esting applications via the parameterized PicardndashVessiot (PPV) theory to questions of integrabil-
ity [22 43] and hypertranscendence [14 24] support a more detailed study of the linear case
Received April 5 2013 Revised November 29 2013 Accepted December 2 2013
copy The Author 2014 Published by Oxford University Press All rights reserved For permissions
please e-mail journalspermissionsoupcom
2 A Minchenko et al
Although there are several similarities between the theory of LDAGs and the theory of linear
algebraic groups (LAGs) a major difference lies in the representation theory of reductive groups
If G is a reductive LAG defined over a field of characteristic 0 then any representation of G is
completely reducible that is any invariant subspace has an invariant complement This is no
longer the case for reductive LDAGs For example if k is a differential field containing at least one
element whose derivative is nonzero the reductive LDAG SL2(k) has a representation in SL4(k)
given by
A 7rarr(
A Aprime
0 A
)
One can show that this is not completely reducible (cf Example 62) Examples such as this show
that the process of taking derivatives complicates the representation theory in a significant way
Initial steps to understand representations of LDAGs are given in [8 9] and a classification of
semisimple LDAGs is given in [13] A Tannakian approach to the representation theory of LDAGs
was introduced in [44 45] (see also [29 28]) and successfully used to further our understanding
of representations of reductive LDAGs in [39 40] This Tannakian approach gives a powerful tool
in which one can understand the impact of taking derivatives on the representation theory of
LDAGs
The main results of the paper consist of bounds for orders of derivatives in differential rep-
resentations of semisimple and reductive LDAGs (Theorems 45 and 49 respectively) Simplified
our results say that for a semisimple LDAG the orders of derivatives are bounded by the dimen-
sion of the representation For a reductive LDAG containing a finitely generated group dense
in the Kolchin topology (cf Section 2) they are bounded by the maximum of the bound for its
semisimple part and by the order of differential equations that define the torus of the group
This result completes and substantially extends what could be proved using [40] where one is
restricted just to SL2 one derivation and to those representations that are extensions of just two
irreducible representations We expect that the main results of the present paper will be used in
the future to give a complete classification of differential representations of semisimple LDAGs
(as this was partially done for SL2 in [40]) Although reductive and semisimple differential alge-
braic groups were studied in [13 39] the techniques used there were not developed enough to
achieve the goals of this paper The main technical tools that we develop and use in our paper
are filtrations of modules of reductive LDAGs which as we show coincide with socle filtrations
in the semisimple case (cf [4 31]) We expect that this technique is general and powerful enough
to have applications beyond this paper
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 3
In this paper we also apply these results to the Galois theory of parameterized linear differ-
ential equations The classical differential Galois theory studies symmetry groups of solutions of
linear differential equations or equivalently the groups of automorphisms of the corresponding
extensions of differential fields The groups that arise are LAGs over the field of constants This
theory started in the 19th century by Picard and Vessiot was put on a firm modern footing by
Kolchin [32] A generalized differential Galois theory that uses Kolchinrsquos axiomatic approach [34]
and realizes differential algebraic groups as Galois groups was initiated in [36]
The PPV Galois theory considered by Cassidy and Singer in [14] is a special case of the
Landesman generalized differential Galois theory and studies symmetry groups of the solutions of
linear differential equations whose coefficients contain parameters This is done by constructing
a differential field containing the solutions and their derivatives with respect to the parameters
called a PPV extension and studying its group of differential symmetries called a parameterized
differential Galois group The Galois groups that arise are LDAGs which are defined by polynomial
differential equations in the parameters Another approach to the Galois theory of systems of
linear differential equations with parameters is given in [7] where the authors study Galois groups
for generic values of the parameters It was shown in [19 43] that a necessary and sufficient
condition that an LDAG G is a PPV-Galois group over the field C (x) is that G contains a finitely
generated Kolchin-dense subgroup (under some further restrictions on C )
In Section 5 we show how our main result yields algorithms in the PPV theory For systems
of differential equations without parameters in the usual PicardndashVessiot theory there are many
existing algorithms for computing differential Galois groups A complete algorithm over the field
C (x) where C is a computable algebraically closed field of constants x is transcendental over
C and its derivative is equal to 1 is given in [58] (see also [15] for the case when the group is
reductive) More efficient algorithms for equations of low order appear in [35 51 52 53 56 57]
These latter algorithms depend on knowing a list of groups that can possibly occur and step-by-
step eliminating the choices
For parameterized systems the first known algorithms are given in [1 18] which apply
to systems of first and second orders (see also [2] for the application of these techniques
to the incomplete gamma function) An algorithm for the case in which the quotient of the
parameterized Galois group by its unipotent radical is constant is given in [41] In the present
paper without any restrictions to the order of the equations based on our main result (upper
bounds mentioned above) we present algorithms that
1 compute the quotient of the parameterized Galois group G by its unipotent radical Ru(G)
2 test whether G is reductive (ie whether Ru(G) = id)
4 A Minchenko et al
Note that these algorithms imply that we can determine if the PPV-Galois group is reductive and
if it is compute it
The paper is organized as follows We start by recalling the basic definitions of differential
algebra differential dimension differential algebraic groups their representations and unipo-
tent and reductive differential algebraic groups in Section 2 The main technical tools of the paper
properties of LDAGs containing a Kolchin-dense finitely generated subgroup and grading filtra-
tions of differential coordinate rings can be found in Sections 223 and 3 respectively The main
result is in Section 4 The main algorithms are described in Section 5 Examples that show that the
main upper bound is sharp and illustrate the algorithm are in Section 6
2 Basic definitions
21 Differential algebra
We begin by fixing notation and recalling some basic facts from differential algebra (cf [33])
In this paper a ∆-ring will be a commutative associative ring R with unit 1 and commuting
derivations ∆= part1 partm We let
Θ = part
i11 middot middotpartim
m | i j Ecirc 0
and note that this free semigroup acts naturally on R For an element parti11 middot middotpartim
m isinΘ we let
ord(part
i11 middot middotpartim
m)
= i1 + + im
Let Y = y1 yn be a set of variables and
ΘY = θy j |θ isinΘ 1 Eacute j Eacute n
The ring of differential polynomials RY in differential indeterminates Y over R is R[ΘY ] with
the derivations parti that extends the parti -action on R as follows
parti(θy j
)= (parti middotθ)y j 1 Eacute j Eacute n 1 Eacute i Eacute m
An ideal I in a∆-ring R is called a differential ideal if parti (a) isin I for all a isin I 1 Eacute i Eacute m For F sub R [F ]
denotes the differential ideal of R generated by F
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 5
Let K be a ∆-field of characteristic zero We denote the subfield of constants of K by
K∆ = c isin K | parti (c) = 0 1 Eacute i Eacute m
Let U be a differentially closed field containing K that is a ∆- extension field of K such that any
system of polynomial differential equations with coefficients in U having a solution in some ∆-
extension of U already have a solution in U n (see [14 Definition 32] and the references therein)
Definition 21 A Kolchin-closed subset W (U ) of U n over K is the set of common zeroes of a
system of differential algebraic equations with coefficients in K that is for f1 fl isin KY we
define
W (U ) = a isinU n | f1(a) = = fl (a) = 0
If W (U ) is a Kolchin-closed subset of U n over K we let I(W ) = f isin Ky1 yn | f (w) = 0 forall w isinW (U )
One has the usual correspondence between Kolchin-closed subsets of Kn defined over K and
radical differential ideals of Ky1 yn Given a Kolchin-closed subset W of U n defined over K
we let the coordinate ring KW be defined as
KW = Ky1 ynI(W )
A differential polynomial map ϕ W1 rarr W2 between Kolchin-closed subsets of U n1 and U n2
respectively defined over K is given in coordinates by differential polynomials in KW1 More-
over to give ϕ W1 rarr W2 is equivalent to defining a differential K-homomorphism ϕlowast KW2 rarrKW1 If KW is an integral domain then W is called irreducible This is equivalent to I(W ) being
a prime differential ideal More generally if
I(W ) = p1 cap cappq
is a minimal prime decomposition which is unique up to permutation [30 VII29] then the
irreducible Kolchin-closed sets W1 Wq corresponding to p1 pq are called the irreducible
components of W We then have
W =W1 cup cupWq
If W is an irreducible Kolchin-closed subset of U n defined over K we denote the quotient field of
KW by KlangW rang
6 A Minchenko et al
In the following we shall need the notion of a Kolchin closed set being of differential type at
most zero The general concept of differential type is defined in terms of the Kolchin polynomial
([33 Section II12]) but this more restricted notion has a simpler definition
Definition 22 Let W be an irreducible Kolchin-closed subset of U n defined over K We say that
W is of differential type at most zero and denote this by τ(W ) Eacute 0 if tr degKKlangWrang ltinfin If W is an
arbitrary Kolchin-closed subset of U n defined over K we say that W has differential type at most
zero if this is true for each of its components
We shall use the fact that if H E G are LDAGs then τ(H) Eacute 0 and τ(GH) Eacute 0 if and only if
τ(G) Eacute 0 [34 Section IV4]
22 Linear Differential Algebraic Groups
Let K subU be as above Recall that LDAG stands for linear differential algebraic group
Definition 23 [8 Chapter II Section 1 p 905] An LDAG over K is a Kolchin-closed subgroup G
of GLn(U ) over K that is an intersection of a Kolchin-closed subset of U n2with GLn(U ) that is
closed under the group operations
Note that we identify GLn(U ) with a Zariski closed subset of U n2+1 given by
(A a) | (det(A)) middota minus1 = 0
If X is an invertible n timesn matrix we can identify it with the pair (X 1det(X )) Hence we may
represent the coordinate ring of GLn(U ) as KX 1det(X ) As usual let Gm(U ) and Ga(U ) denote
the multiplicative and additive groups of U respectively The coordinate ring of the LDAG SL2(U )
is isomorphic to
Kc11c12c21c22[c11c22 minus c12c21 minus1]
For a group G sub GLn(U ) we denote the Zariski closure of G in GLn(U ) by G Then G is a LAG over
U If G sub GLn(U ) is an LDAG defined over K then G is defined over K as well
The irreducible component of an LDAG G containing id the identity is called the identity
component of G and denoted by G An LDAG G is called connected if G =G which is equivalent
to G being an irreducible Kolchin closed set [8 p 906]
The coordinate ring KG of an LDAG G has a structure of a differential Hopf algebra that is
a Hopf algebra in which the coproduct antipode and counit are homomorphisms of differential
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 7
algebras [44 Section 32] and [9 Section 2] One can view G as a representable functor defined
on K-algebras represented by KG For example if V is an n-dimensional vector space over K
GL(V ) = AutV is an LDAG represented by KGLn = KGLn(U )
221 Representations of LDAGs
Definition 24 [9][44 Definition 6] Let G be an LDAG A differential polynomial group homo-
morphism
rV G rarr GL(V )
is called a differential representation of G where V is a finite-dimensional vector space over K
Such space is simply called a G-module This is equivalent to giving a comodule structure
ρV V rarrV otimesK KG
see [44 Definition 7 and Theorem 1] [59 Section 32] Moreover if U sub V is a submodule then
V |U = U
As usual morphisms between G-modules are K-linear maps that are G-equivariant The
category of differential representations of G is denoted by RepG
For an LDAG G let A = KG be its differential Hopf algebra and
∆ A rarr AotimesK A
be the comultiplication inducing the right-regular G-module structure on A as follows (see also
[44 Section 41]) For g x isinG(U ) and f isin A
(rg ( f )
)(x) = f (x middot g ) =∆( f )(x g ) =
nsumi=1
fi (x)gi (g )
where ∆( f ) =sumni=1 fi otimes gi The k-vector space A is an A-comodule via
A =∆
Proposition 25 [59 Corollary 33 Lemma 35][44 Lemma 3] The coalgebra A is a countable
union of its finite-dimensional subcoalgebras If V isin RepG then as an A-comodule V embeds
into AdimV
8 A Minchenko et al
By [8 Proposition 7] ρ(G) sub GL(V ) is a differential algebraic subgroup Given a representa-
tion ρ of an LDAG G one can define its prolongations
Pi (ρ) G rarr GL(Pi (V ))
with respect to parti as follows (see [21 Section 52] [44 Definition 4 and Theorem 1] and [39
p 1199]) Let
Pi (V ) = K ((KoplusKparti )K otimesK V ) (21)
as vector spaces where KoplusKparti is considered as the right K-module parti middota = parti (a)+aparti for all a isin K
Then the action of G is given by Pi (ρ) as follows
Pi (ρ)(g )(1otimes v) = 1otimesρ(g )(v) Pi (ρ)(g )(parti otimes v) = parti otimesρ(g )(v)
for all g isinG and v isinV In the language of matrices if Ag isin GLn corresponds to the action of g isinG
on V then the matrix (Ag parti Ag
0 Ag
)
corresponds to the action of g on Pi (V ) In what follows the q th iterate of Pi is denoted by P qi
Moreover the above induces the exact sequences
0 minusminusminusminusminusrarr Vιiminusminusminusminusminusrarr Pi (V )
πiminusminusminusminusminusrarr V minusminusminusminusminusrarr 0 (22)
where ιi (v) = 1otimesv and πi (aotimesu+bparti otimesv) = bv u v isinV a b isin K For any integer s we will refer to
P smP s
mminus1 middot middotP s1(ρ) G rarrGLNs
to be the s th total prolongation of ρ (where Ns is the dimension of the underlying prolonged
vector space) We denote this representation by P s (ρ) G rarr GLNs The underlying vector space
is denoted by P s (V )
It will be convenient to consider A as a G-module For this let RepG denote the differential
tensor category of all A-comodules (not necessarily finite-dimensional) which are direct limits of
finite-dimensional A-comodules by [59 Section 33] Then A isin RepG by Proposition 25
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 9
222 Unipotent radical of differential algebraic groups and reductive LDAGs
Definition 26 [10 Theorem 2] Let G be an LDAG defined over K We say that G is unipotent if
one of the following conditions holds
1 G is conjugate to a differential algebraic subgroup of the group Un of unipotent upper
triangular matrices
2 G contains no elements of finite order gt 1
3 G has a descending normal sequence of differential algebraic subgroups
G =G0 supG1 sup supGN = 1
with Gi Gi+1 isomorphic to a differential algebraic subgroup of the additive group Ga
One can show that an LDAG G defined over K admits a maximal normal unipotent differen-
tial subgroup [39 Theorem 310]
Definition 27 This subgroup is called the unipotent radical of G and denoted by Ru(G) The
unipotent radical of a LAG H is also denoted by Ru(H)
Definition 28 [39 Definition 312] An LDAG G is called reductive if its unipotent radical is trivial
that is Ru(G) = id
Remark 29 If G is given as a linear differential algebraic subgroup of some GLν we may consider
its Zariski closure G in GLν which is an algebraic group scheme defined over K Then following
the proof of [39 Theorem 310]
Ru(G) = Ru
(G
)capG
This implies that if G is reductive then G is reductive However in general the Zariski closure of
Ru(G) may be strictly included in Ru(G) [39 Ex 317]
223 Differentially finitely generated groups
As mentioned in the introduction one motivation for studying LDAGs is their use in the PPV
theory In Section 5 we will discuss PPV-extensions of certain fields whose PPV-Galois groups
satisfy the following property In this subsection we will assume that K is differentially closed
Definition 210 Let G be an LDAG defined over K We say that G is differentially finitely generated
or simply a DFGG if G(K) contains a finitely generated subgroup that is Kolchin dense over K
10 A Minchenko et al
Proposition 211 If G is a DFGG then its identity component G is a DFGG
Proof The ReidemeisterndashSchreier Theorem implies that a subgroup of finite index in a finitely
generated group is finitely generated ([38 Corollary 271]) One can use this fact to construct a
proof of the above Nonetheless we present a self-contained proof
Let F = GG and t = |GG| We claim that every sequence of t elements of F has a
contiguous subsequence whose product is the identity To see this let a1 at be a sequence
of elements of F Set
b1 = a1b2 = a1a2 bt = a1a2 middot middotat
If there are i lt j such that bi = b j then
id = bminus1i b j = a j+1 middot middota j
If the b j are pairwise distinct they exhaust F and so one of them must be the identity
Let S = Sminus1 be a finite set generating a dense subgroup ΓsubG Set
Γ0 = s | s = s1 middot middot sm isinG si isin S
Then Γ0 is a Kolchin dense subgroup of G Applying the above observation concerning F we see
that Γ0 is generated by the finite set
S0 = s | s = s1 middot middot sm isinG si isin S and m Eacute |GG|
Lemma 212 If H sub Gma is a DFGG then τ(H) Eacute 0
Proof Let πi be the projection of Gma onto its i th factor We have that πi (H) sub Ga is a DFGG and
so by [41 Lemma 210] τ(πi (H)) Eacute 0 Since
H subπ1(H)times timesπm(H) and τ(π1(H)times timesπm(H)) Eacute 0
we have τ(H) = 0
Lemma 213 If H sub Grm is a DFGG then τ(H) Eacute 0
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 11
Proof Let `∆ Grm rarr Gr m
a be the homomorphism
`∆(y1 yr ) =(part1 y1
y1
part1 yr
yrpart2 y1
y1
part2 yr
yr
partm y1
y1
partm yr
yr
)
The image of H under this homomorphism is a DFGG in Gr ma and so has differential type at most
0 The kernel of this homomorphism restricted to H is
(Gm
(K∆
))r capH
which also has type at most 0 Therefore τ(H) Eacute 0
Lemma 214 Let G be a reductive LDAG Then G is a DFGG if and only if τ(Z (G)
)Eacute 0
Proof Assume that G is a DFGG By Proposition 211 we can assume that G is Kolchin-connected
as well as a DFGG From [39 Theorem 47] we can assume that G = P is a reductive LAG From
the structure of reductive LAGs we know that
P = (PP ) middotZ (P )
where Z (P ) denotes the center (PP ) is the commutator subgroup and Z (P )cap (PP ) is finite Note
also that Z (P ) is a torus and that Z (G) = Z (P )capG Let
π P rarr P(PP ) Z (P )[Z (P )cap (PP )]
The image of G is connected and so lies in
π(Z (P )
) Gtm
for some t The image is a DFGG and so by Lemma 213 must have type at most 0 From the
description of π one sees that
π Z (G) rarr Z (G)[Z (P )cap (PP )] sub Z (P )[Z (P )cap (PP )]
Since Z (P )cap (PP ) is finite we have τ(Z (G)
)Eacute 0
Nowadays assume that τ(Z (G)
) Eacute 0 [41 Proposition 29] implies that Z (G) is a DFGG
Therefore it is enough to show that G prime = GZ (G) is a DFGG We see that G prime is semisimple and
12 A Minchenko et al
we will show that any semisimple LDAG is a DFGG Clearly it is enough to show that this is true
under the further assumption that G prime is connected
Let D be the K-vector space spanned by ∆ [13 Theorem 18] implies that G prime = G1 middot middotG`
where for each i there exists a simple LAG Hi defined over Q and a Lie (A Lie subspace E subD is
a subspace such that for any partpartprime isin E we have partpartprimeminuspartprimepart isin E ) K -subspace Ei of D such that
Gi = Hi
(KEi
) KEi =
c isin K | part(c) = 0 for all part isin Ei
Therefore it suffices to show that for a simple LAG H and a Lie K-subspace E sub D the LDAG
H(KE
)is a DFGG From [34 Proposition 6 and 7] E has a K-basis of commuting derivations Λ=
partprime1 partprimer which can be extended to a commuting basis
partprime1 partprimem
of D Let Π=
partprimer+1 partprimem
[14 Lemma 93] implies that KE is differentially closed as a Π-differential field We may consider
H(KE
)as a LAG over theΠ-differential field KE The result now follows from [50 Lemma 22]
3 Filtrations and gradings of the coordinate ring of an LDAG
In this section we develop the main technique of the paper filtrations and grading of coordinate
rings of LDAGs Let K be a ∆-field of characteristic zero not necessarily differentially closed The
set of natural numbers 012 is denoted byN
31 Filtrations of G-modules
Let G be an LDAG and A = KG be the corresponding differential Hopf algebra (see [9 Section 2]
and [44 Section 32]) Fix a faithful G-module W Let
ϕ KGL(W ) rarr A (31)
be the differential epimorphism of differential Hopf algebras corresponding to the embedding
G rarr GL(W ) Set
H =G
which is a LAG Define
A0 =ϕ(K[GL(W )]) = K[H ] (32)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 13
and for n Ecirc 1
An = spanK
prodjisinJ
θ j y j isin A∣∣∣ J is a finite set y j isin A0 θ j isinΘ
sumjisinJ
ord(θ j ) Eacute n
(33)
The following shows that the subspaces An sub A form a filtration (in the sense of [55]) of the
Hopf algebra A
Proposition 31 We have
A = ⋃nisinN
An An sub An+1 (34)
Ai A j sub Ai+ j i j isinN (35)
∆(An) subnsum
i=0Ai otimesK Anminusi (36)
Proof Relation (35) follows immediately from (33) Since K[GL(W )] differentially generates
KGL(W ) and ϕ is a differential epimorphism A0 differentially generates A which implies (34)
Finally let us prove (36) Consider the differential Hopf algebra
B = AotimesK A
where partl 1 Eacute l Eacute m acts on B as follows
partl (x otimes y) = partl (x)otimes y +x otimespartl (y) x y isin A
Set
Bn =nsum
i=0Ai otimesK Anminusi n isinN
We have
Bi B j sub Bi+ j and partl (Bn) sub Bn+1 i j isinN n isinN 1 Eacute l Eacute m (37)
Since K[GL(W )] is a Hopf subalgebra of KGL(W ) A0 is a Hopf subalgebra of A In particular
∆(A0) sub B0 (38)
14 A Minchenko et al
Since ∆ A rarr B is a differential homomorphism definition (33) and relations (38) (37) imply
∆(An) sub Bn n isinN
We will call AnnisinN the W -filtration of A As the definition of An depends on W we will
sometimes write An(W ) for An By (36) An is a subcomodule of A If x isin A An then the relation
x = (εotimes Id)∆(x) (39)
shows that ∆(x) 6isin A otimes An Therefore An is the largest subcomodule U sub A such that ∆(U ) subU otimesK An This suggests the following notation
For V isin RepG and n isinN let Vn denote the largest submodule U subV such that
V (U ) subU otimesK An
Then submodules Vn subV n isinN form a filtration of V which we also call the W -filtration
Proposition 32 For a morphism f U rarrV of G-modules and an n isinN we have f (Un) subVn
Proof The proof follows immediately from the definition of a morphism of G-modules
Note that Un subVn and Vn capU subUn for all submodules U subV isin RepG Therefore
Un =U capVn for every subcomodule U subV isin RepG (310)
(U oplusV )n =Un oplusVn for all U V isin RepG (311)(⋃iisinNV (i )
)n =⋃
iisinNV (i )n V (i ) subV (i +1) isin RepG (312)
Proposition 33 For every V isin RepG we have
V (Vn) subnsum
i=0Vi otimesK Anminusi (313)
Proof Let X denote the set of all V isin RepG satisfying (313) It follows from (310) and (311) that
if U V isin X then every submodule of U oplusV belongs to X If V isin RepG then V is isomorphic to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 15
submodule of AdimV by Proposition 25 Since A isin X by Proposition 31 Ob(RepG) sub X For the
general case it remains to apply (312)
Recall that a module is called semisimple if it equals the sum of its simple submodules
Proposition 34 Suppose that W is a semisimple G-module Then the LAG H is reductive If W
is not semisimple then it is not semisimple as an H-module
Proof For the proof see [39 proof of Theorem 47]
Lemma 35 Let V isin RepG If V is semisimple then V = V0 (Loosely speaking this means that
all completely reducible representations of an LDAG are polynomial This was also proved in [39
Theorem 33]) If W is semisimple the converse is true
Proof By (311) it suffices to prove the statement for a simple V isin RepG Suppose that V is
simple and V =Vn 6=Vnminus1 Then Vnminus1 = 0 and Proposition 33 implies
V (V ) subV otimes A0 (314)
Hence V =V0
Suppose that W is semisimple and V = V0 isin RepG The latter means (314) that is the
representation of G on V extends to the representation of H on V But H is reductive by
Proposition 34 (since W is semisimple) Then V is semisimple as an H-module Again by
Proposition 34 the G-module V is semisimple
Corollary 36 If W is semisimple then A0 is the sum of all simple subcomodules of A Therefore
if U V are faithful semisimple G-modules then the U - and V -filtrations of A coincide
Proof By Lemma 35 if Z sub A is simple then Z = Z0 Hence by Proposition 32 Z is contained
in A0 Moreover by Lemma 35 A0 is the sum of all its simple submodules
Corollary 37 The LDAG G is connected if and only if the LAG H is connected
Proof If G is Kolchin connected and
A = KG = KGL(W )p= KXi j 1detp
16 A Minchenko et al
then the differential ideal p is prime [8 p 895] Since by [8 p 897]
A0 = K[H ] = K[GL(W )]
(pcapK[GL(W )]) = K[Xi j 1det]
(pcapK[Xi j 1det])
and the ideal pcapK[Xi j 1det] is prime H is Zariski connected
Set Γ =GG which is finite Denote the quotient map by
π G rarr Γ
Since Γ is finite and charK = 0 B = KΓ isin RepΓ is semisimple Then B has a structure of a
semisimple G-module via π Therefore by Lemma 35 B = B0 Since πlowast is a homomorphism of
G-modules by Proposition 32
πlowast(B) =πlowast(B0) sub A0 = K[H ]
This means that π is a restriction of an epimorphism H rarr Γ which completes the proof
For the ∆-field K denote the underlying abstract field endowed with the trivial differential
structure (partl k = 0 1 Eacute l Eacute m) by K
Proposition 38 Suppose that the LDAG G is connected If x isin Ai y isin A j and x y isin Ai+ jminus1 then
either x isin Aiminus1 or y isin A jminus1
Proof We need to show that the graded algebra
gr A = oplusnisinN
AnAnminus1
is an integral domain Note that gr A is a differential algebra via
partl (x + Anminus1) = partl (x)+ An x isin An
Furthermore to a homomorphism ν B rarrC of filtered algebras such that ν(Bn) subCn n isinN there
corresponds the homomorphism
grν grB rarr grC x +Bnminus1 7rarr ν(x)+Cnminus1 x isin Bn
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 17
Let us identify GL(W ) with GLd d = dimW and set
B =Qxi j 1det
the coordinate ring of GLd overQ The algebra B is graded by
B n = spanQ
prodjisinJ
θ j y j
∣∣∣ J is a finite set y j isinQ[GLd ] θ j isinΘsumjisinJ
ord(θ j ) = n
n isinN
The W -filtration of B is then associated with this grading
Bn =noplus
i=0B i
For a field extension Qsub L set LB = B otimesQ L a Hopf algebra over L Then the algebra LB is graded
by LB n = B n otimesL
Let I stand for the Hopf ideal of KB defining G sub GLd For x isin KB let xh denote the highest
degree component of x with respect to the grading
KB n Let I denote the K-span of all xh x isin I
As in the proof of Proposition 31 we conclude that for all n isinN
∆(B n
)sub nsumi=0
B i otimesK B nminusi (315)
Since ∆(I ) sub I otimesK B +B otimesK I inclusion (315) implies that for all n isinN and x isin I capBn
I otimesK Bn +Bn otimesK I 3∆(x) =∆(x minusxh)+∆(xh) isin(
nminus1sumi=0
Bi otimesK Bnminusiminus1
)oplus
(nsum
i=0B i otimesK B nminusi
)
Hence by induction one has
∆(xh) isin I otimesK Bn +Bn otimesK I sub I otimesK B +B otimesK I
We have S(I ) sub I where S B rarr B is the antipode Moreover since S(B0) = B0 and S is differential
S(B n
)sub B n n isinN
Hence
S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n
18 A Minchenko et al
which implies that
S(I)sub I
Therefore I is a Hopf ideal of KB (not necessarily differential) Consider the algebra map
α KBβ grK B
grϕminusrarr gr A
where β is defined by the sections
KB n rarr KBn
KBnminus1 n isinN
and ϕ is given by (31) For every x isin I let n isinN be such that xh isin B n Then
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
2 A Minchenko et al
Although there are several similarities between the theory of LDAGs and the theory of linear
algebraic groups (LAGs) a major difference lies in the representation theory of reductive groups
If G is a reductive LAG defined over a field of characteristic 0 then any representation of G is
completely reducible that is any invariant subspace has an invariant complement This is no
longer the case for reductive LDAGs For example if k is a differential field containing at least one
element whose derivative is nonzero the reductive LDAG SL2(k) has a representation in SL4(k)
given by
A 7rarr(
A Aprime
0 A
)
One can show that this is not completely reducible (cf Example 62) Examples such as this show
that the process of taking derivatives complicates the representation theory in a significant way
Initial steps to understand representations of LDAGs are given in [8 9] and a classification of
semisimple LDAGs is given in [13] A Tannakian approach to the representation theory of LDAGs
was introduced in [44 45] (see also [29 28]) and successfully used to further our understanding
of representations of reductive LDAGs in [39 40] This Tannakian approach gives a powerful tool
in which one can understand the impact of taking derivatives on the representation theory of
LDAGs
The main results of the paper consist of bounds for orders of derivatives in differential rep-
resentations of semisimple and reductive LDAGs (Theorems 45 and 49 respectively) Simplified
our results say that for a semisimple LDAG the orders of derivatives are bounded by the dimen-
sion of the representation For a reductive LDAG containing a finitely generated group dense
in the Kolchin topology (cf Section 2) they are bounded by the maximum of the bound for its
semisimple part and by the order of differential equations that define the torus of the group
This result completes and substantially extends what could be proved using [40] where one is
restricted just to SL2 one derivation and to those representations that are extensions of just two
irreducible representations We expect that the main results of the present paper will be used in
the future to give a complete classification of differential representations of semisimple LDAGs
(as this was partially done for SL2 in [40]) Although reductive and semisimple differential alge-
braic groups were studied in [13 39] the techniques used there were not developed enough to
achieve the goals of this paper The main technical tools that we develop and use in our paper
are filtrations of modules of reductive LDAGs which as we show coincide with socle filtrations
in the semisimple case (cf [4 31]) We expect that this technique is general and powerful enough
to have applications beyond this paper
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 3
In this paper we also apply these results to the Galois theory of parameterized linear differ-
ential equations The classical differential Galois theory studies symmetry groups of solutions of
linear differential equations or equivalently the groups of automorphisms of the corresponding
extensions of differential fields The groups that arise are LAGs over the field of constants This
theory started in the 19th century by Picard and Vessiot was put on a firm modern footing by
Kolchin [32] A generalized differential Galois theory that uses Kolchinrsquos axiomatic approach [34]
and realizes differential algebraic groups as Galois groups was initiated in [36]
The PPV Galois theory considered by Cassidy and Singer in [14] is a special case of the
Landesman generalized differential Galois theory and studies symmetry groups of the solutions of
linear differential equations whose coefficients contain parameters This is done by constructing
a differential field containing the solutions and their derivatives with respect to the parameters
called a PPV extension and studying its group of differential symmetries called a parameterized
differential Galois group The Galois groups that arise are LDAGs which are defined by polynomial
differential equations in the parameters Another approach to the Galois theory of systems of
linear differential equations with parameters is given in [7] where the authors study Galois groups
for generic values of the parameters It was shown in [19 43] that a necessary and sufficient
condition that an LDAG G is a PPV-Galois group over the field C (x) is that G contains a finitely
generated Kolchin-dense subgroup (under some further restrictions on C )
In Section 5 we show how our main result yields algorithms in the PPV theory For systems
of differential equations without parameters in the usual PicardndashVessiot theory there are many
existing algorithms for computing differential Galois groups A complete algorithm over the field
C (x) where C is a computable algebraically closed field of constants x is transcendental over
C and its derivative is equal to 1 is given in [58] (see also [15] for the case when the group is
reductive) More efficient algorithms for equations of low order appear in [35 51 52 53 56 57]
These latter algorithms depend on knowing a list of groups that can possibly occur and step-by-
step eliminating the choices
For parameterized systems the first known algorithms are given in [1 18] which apply
to systems of first and second orders (see also [2] for the application of these techniques
to the incomplete gamma function) An algorithm for the case in which the quotient of the
parameterized Galois group by its unipotent radical is constant is given in [41] In the present
paper without any restrictions to the order of the equations based on our main result (upper
bounds mentioned above) we present algorithms that
1 compute the quotient of the parameterized Galois group G by its unipotent radical Ru(G)
2 test whether G is reductive (ie whether Ru(G) = id)
4 A Minchenko et al
Note that these algorithms imply that we can determine if the PPV-Galois group is reductive and
if it is compute it
The paper is organized as follows We start by recalling the basic definitions of differential
algebra differential dimension differential algebraic groups their representations and unipo-
tent and reductive differential algebraic groups in Section 2 The main technical tools of the paper
properties of LDAGs containing a Kolchin-dense finitely generated subgroup and grading filtra-
tions of differential coordinate rings can be found in Sections 223 and 3 respectively The main
result is in Section 4 The main algorithms are described in Section 5 Examples that show that the
main upper bound is sharp and illustrate the algorithm are in Section 6
2 Basic definitions
21 Differential algebra
We begin by fixing notation and recalling some basic facts from differential algebra (cf [33])
In this paper a ∆-ring will be a commutative associative ring R with unit 1 and commuting
derivations ∆= part1 partm We let
Θ = part
i11 middot middotpartim
m | i j Ecirc 0
and note that this free semigroup acts naturally on R For an element parti11 middot middotpartim
m isinΘ we let
ord(part
i11 middot middotpartim
m)
= i1 + + im
Let Y = y1 yn be a set of variables and
ΘY = θy j |θ isinΘ 1 Eacute j Eacute n
The ring of differential polynomials RY in differential indeterminates Y over R is R[ΘY ] with
the derivations parti that extends the parti -action on R as follows
parti(θy j
)= (parti middotθ)y j 1 Eacute j Eacute n 1 Eacute i Eacute m
An ideal I in a∆-ring R is called a differential ideal if parti (a) isin I for all a isin I 1 Eacute i Eacute m For F sub R [F ]
denotes the differential ideal of R generated by F
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 5
Let K be a ∆-field of characteristic zero We denote the subfield of constants of K by
K∆ = c isin K | parti (c) = 0 1 Eacute i Eacute m
Let U be a differentially closed field containing K that is a ∆- extension field of K such that any
system of polynomial differential equations with coefficients in U having a solution in some ∆-
extension of U already have a solution in U n (see [14 Definition 32] and the references therein)
Definition 21 A Kolchin-closed subset W (U ) of U n over K is the set of common zeroes of a
system of differential algebraic equations with coefficients in K that is for f1 fl isin KY we
define
W (U ) = a isinU n | f1(a) = = fl (a) = 0
If W (U ) is a Kolchin-closed subset of U n over K we let I(W ) = f isin Ky1 yn | f (w) = 0 forall w isinW (U )
One has the usual correspondence between Kolchin-closed subsets of Kn defined over K and
radical differential ideals of Ky1 yn Given a Kolchin-closed subset W of U n defined over K
we let the coordinate ring KW be defined as
KW = Ky1 ynI(W )
A differential polynomial map ϕ W1 rarr W2 between Kolchin-closed subsets of U n1 and U n2
respectively defined over K is given in coordinates by differential polynomials in KW1 More-
over to give ϕ W1 rarr W2 is equivalent to defining a differential K-homomorphism ϕlowast KW2 rarrKW1 If KW is an integral domain then W is called irreducible This is equivalent to I(W ) being
a prime differential ideal More generally if
I(W ) = p1 cap cappq
is a minimal prime decomposition which is unique up to permutation [30 VII29] then the
irreducible Kolchin-closed sets W1 Wq corresponding to p1 pq are called the irreducible
components of W We then have
W =W1 cup cupWq
If W is an irreducible Kolchin-closed subset of U n defined over K we denote the quotient field of
KW by KlangW rang
6 A Minchenko et al
In the following we shall need the notion of a Kolchin closed set being of differential type at
most zero The general concept of differential type is defined in terms of the Kolchin polynomial
([33 Section II12]) but this more restricted notion has a simpler definition
Definition 22 Let W be an irreducible Kolchin-closed subset of U n defined over K We say that
W is of differential type at most zero and denote this by τ(W ) Eacute 0 if tr degKKlangWrang ltinfin If W is an
arbitrary Kolchin-closed subset of U n defined over K we say that W has differential type at most
zero if this is true for each of its components
We shall use the fact that if H E G are LDAGs then τ(H) Eacute 0 and τ(GH) Eacute 0 if and only if
τ(G) Eacute 0 [34 Section IV4]
22 Linear Differential Algebraic Groups
Let K subU be as above Recall that LDAG stands for linear differential algebraic group
Definition 23 [8 Chapter II Section 1 p 905] An LDAG over K is a Kolchin-closed subgroup G
of GLn(U ) over K that is an intersection of a Kolchin-closed subset of U n2with GLn(U ) that is
closed under the group operations
Note that we identify GLn(U ) with a Zariski closed subset of U n2+1 given by
(A a) | (det(A)) middota minus1 = 0
If X is an invertible n timesn matrix we can identify it with the pair (X 1det(X )) Hence we may
represent the coordinate ring of GLn(U ) as KX 1det(X ) As usual let Gm(U ) and Ga(U ) denote
the multiplicative and additive groups of U respectively The coordinate ring of the LDAG SL2(U )
is isomorphic to
Kc11c12c21c22[c11c22 minus c12c21 minus1]
For a group G sub GLn(U ) we denote the Zariski closure of G in GLn(U ) by G Then G is a LAG over
U If G sub GLn(U ) is an LDAG defined over K then G is defined over K as well
The irreducible component of an LDAG G containing id the identity is called the identity
component of G and denoted by G An LDAG G is called connected if G =G which is equivalent
to G being an irreducible Kolchin closed set [8 p 906]
The coordinate ring KG of an LDAG G has a structure of a differential Hopf algebra that is
a Hopf algebra in which the coproduct antipode and counit are homomorphisms of differential
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 7
algebras [44 Section 32] and [9 Section 2] One can view G as a representable functor defined
on K-algebras represented by KG For example if V is an n-dimensional vector space over K
GL(V ) = AutV is an LDAG represented by KGLn = KGLn(U )
221 Representations of LDAGs
Definition 24 [9][44 Definition 6] Let G be an LDAG A differential polynomial group homo-
morphism
rV G rarr GL(V )
is called a differential representation of G where V is a finite-dimensional vector space over K
Such space is simply called a G-module This is equivalent to giving a comodule structure
ρV V rarrV otimesK KG
see [44 Definition 7 and Theorem 1] [59 Section 32] Moreover if U sub V is a submodule then
V |U = U
As usual morphisms between G-modules are K-linear maps that are G-equivariant The
category of differential representations of G is denoted by RepG
For an LDAG G let A = KG be its differential Hopf algebra and
∆ A rarr AotimesK A
be the comultiplication inducing the right-regular G-module structure on A as follows (see also
[44 Section 41]) For g x isinG(U ) and f isin A
(rg ( f )
)(x) = f (x middot g ) =∆( f )(x g ) =
nsumi=1
fi (x)gi (g )
where ∆( f ) =sumni=1 fi otimes gi The k-vector space A is an A-comodule via
A =∆
Proposition 25 [59 Corollary 33 Lemma 35][44 Lemma 3] The coalgebra A is a countable
union of its finite-dimensional subcoalgebras If V isin RepG then as an A-comodule V embeds
into AdimV
8 A Minchenko et al
By [8 Proposition 7] ρ(G) sub GL(V ) is a differential algebraic subgroup Given a representa-
tion ρ of an LDAG G one can define its prolongations
Pi (ρ) G rarr GL(Pi (V ))
with respect to parti as follows (see [21 Section 52] [44 Definition 4 and Theorem 1] and [39
p 1199]) Let
Pi (V ) = K ((KoplusKparti )K otimesK V ) (21)
as vector spaces where KoplusKparti is considered as the right K-module parti middota = parti (a)+aparti for all a isin K
Then the action of G is given by Pi (ρ) as follows
Pi (ρ)(g )(1otimes v) = 1otimesρ(g )(v) Pi (ρ)(g )(parti otimes v) = parti otimesρ(g )(v)
for all g isinG and v isinV In the language of matrices if Ag isin GLn corresponds to the action of g isinG
on V then the matrix (Ag parti Ag
0 Ag
)
corresponds to the action of g on Pi (V ) In what follows the q th iterate of Pi is denoted by P qi
Moreover the above induces the exact sequences
0 minusminusminusminusminusrarr Vιiminusminusminusminusminusrarr Pi (V )
πiminusminusminusminusminusrarr V minusminusminusminusminusrarr 0 (22)
where ιi (v) = 1otimesv and πi (aotimesu+bparti otimesv) = bv u v isinV a b isin K For any integer s we will refer to
P smP s
mminus1 middot middotP s1(ρ) G rarrGLNs
to be the s th total prolongation of ρ (where Ns is the dimension of the underlying prolonged
vector space) We denote this representation by P s (ρ) G rarr GLNs The underlying vector space
is denoted by P s (V )
It will be convenient to consider A as a G-module For this let RepG denote the differential
tensor category of all A-comodules (not necessarily finite-dimensional) which are direct limits of
finite-dimensional A-comodules by [59 Section 33] Then A isin RepG by Proposition 25
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 9
222 Unipotent radical of differential algebraic groups and reductive LDAGs
Definition 26 [10 Theorem 2] Let G be an LDAG defined over K We say that G is unipotent if
one of the following conditions holds
1 G is conjugate to a differential algebraic subgroup of the group Un of unipotent upper
triangular matrices
2 G contains no elements of finite order gt 1
3 G has a descending normal sequence of differential algebraic subgroups
G =G0 supG1 sup supGN = 1
with Gi Gi+1 isomorphic to a differential algebraic subgroup of the additive group Ga
One can show that an LDAG G defined over K admits a maximal normal unipotent differen-
tial subgroup [39 Theorem 310]
Definition 27 This subgroup is called the unipotent radical of G and denoted by Ru(G) The
unipotent radical of a LAG H is also denoted by Ru(H)
Definition 28 [39 Definition 312] An LDAG G is called reductive if its unipotent radical is trivial
that is Ru(G) = id
Remark 29 If G is given as a linear differential algebraic subgroup of some GLν we may consider
its Zariski closure G in GLν which is an algebraic group scheme defined over K Then following
the proof of [39 Theorem 310]
Ru(G) = Ru
(G
)capG
This implies that if G is reductive then G is reductive However in general the Zariski closure of
Ru(G) may be strictly included in Ru(G) [39 Ex 317]
223 Differentially finitely generated groups
As mentioned in the introduction one motivation for studying LDAGs is their use in the PPV
theory In Section 5 we will discuss PPV-extensions of certain fields whose PPV-Galois groups
satisfy the following property In this subsection we will assume that K is differentially closed
Definition 210 Let G be an LDAG defined over K We say that G is differentially finitely generated
or simply a DFGG if G(K) contains a finitely generated subgroup that is Kolchin dense over K
10 A Minchenko et al
Proposition 211 If G is a DFGG then its identity component G is a DFGG
Proof The ReidemeisterndashSchreier Theorem implies that a subgroup of finite index in a finitely
generated group is finitely generated ([38 Corollary 271]) One can use this fact to construct a
proof of the above Nonetheless we present a self-contained proof
Let F = GG and t = |GG| We claim that every sequence of t elements of F has a
contiguous subsequence whose product is the identity To see this let a1 at be a sequence
of elements of F Set
b1 = a1b2 = a1a2 bt = a1a2 middot middotat
If there are i lt j such that bi = b j then
id = bminus1i b j = a j+1 middot middota j
If the b j are pairwise distinct they exhaust F and so one of them must be the identity
Let S = Sminus1 be a finite set generating a dense subgroup ΓsubG Set
Γ0 = s | s = s1 middot middot sm isinG si isin S
Then Γ0 is a Kolchin dense subgroup of G Applying the above observation concerning F we see
that Γ0 is generated by the finite set
S0 = s | s = s1 middot middot sm isinG si isin S and m Eacute |GG|
Lemma 212 If H sub Gma is a DFGG then τ(H) Eacute 0
Proof Let πi be the projection of Gma onto its i th factor We have that πi (H) sub Ga is a DFGG and
so by [41 Lemma 210] τ(πi (H)) Eacute 0 Since
H subπ1(H)times timesπm(H) and τ(π1(H)times timesπm(H)) Eacute 0
we have τ(H) = 0
Lemma 213 If H sub Grm is a DFGG then τ(H) Eacute 0
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 11
Proof Let `∆ Grm rarr Gr m
a be the homomorphism
`∆(y1 yr ) =(part1 y1
y1
part1 yr
yrpart2 y1
y1
part2 yr
yr
partm y1
y1
partm yr
yr
)
The image of H under this homomorphism is a DFGG in Gr ma and so has differential type at most
0 The kernel of this homomorphism restricted to H is
(Gm
(K∆
))r capH
which also has type at most 0 Therefore τ(H) Eacute 0
Lemma 214 Let G be a reductive LDAG Then G is a DFGG if and only if τ(Z (G)
)Eacute 0
Proof Assume that G is a DFGG By Proposition 211 we can assume that G is Kolchin-connected
as well as a DFGG From [39 Theorem 47] we can assume that G = P is a reductive LAG From
the structure of reductive LAGs we know that
P = (PP ) middotZ (P )
where Z (P ) denotes the center (PP ) is the commutator subgroup and Z (P )cap (PP ) is finite Note
also that Z (P ) is a torus and that Z (G) = Z (P )capG Let
π P rarr P(PP ) Z (P )[Z (P )cap (PP )]
The image of G is connected and so lies in
π(Z (P )
) Gtm
for some t The image is a DFGG and so by Lemma 213 must have type at most 0 From the
description of π one sees that
π Z (G) rarr Z (G)[Z (P )cap (PP )] sub Z (P )[Z (P )cap (PP )]
Since Z (P )cap (PP ) is finite we have τ(Z (G)
)Eacute 0
Nowadays assume that τ(Z (G)
) Eacute 0 [41 Proposition 29] implies that Z (G) is a DFGG
Therefore it is enough to show that G prime = GZ (G) is a DFGG We see that G prime is semisimple and
12 A Minchenko et al
we will show that any semisimple LDAG is a DFGG Clearly it is enough to show that this is true
under the further assumption that G prime is connected
Let D be the K-vector space spanned by ∆ [13 Theorem 18] implies that G prime = G1 middot middotG`
where for each i there exists a simple LAG Hi defined over Q and a Lie (A Lie subspace E subD is
a subspace such that for any partpartprime isin E we have partpartprimeminuspartprimepart isin E ) K -subspace Ei of D such that
Gi = Hi
(KEi
) KEi =
c isin K | part(c) = 0 for all part isin Ei
Therefore it suffices to show that for a simple LAG H and a Lie K-subspace E sub D the LDAG
H(KE
)is a DFGG From [34 Proposition 6 and 7] E has a K-basis of commuting derivations Λ=
partprime1 partprimer which can be extended to a commuting basis
partprime1 partprimem
of D Let Π=
partprimer+1 partprimem
[14 Lemma 93] implies that KE is differentially closed as a Π-differential field We may consider
H(KE
)as a LAG over theΠ-differential field KE The result now follows from [50 Lemma 22]
3 Filtrations and gradings of the coordinate ring of an LDAG
In this section we develop the main technique of the paper filtrations and grading of coordinate
rings of LDAGs Let K be a ∆-field of characteristic zero not necessarily differentially closed The
set of natural numbers 012 is denoted byN
31 Filtrations of G-modules
Let G be an LDAG and A = KG be the corresponding differential Hopf algebra (see [9 Section 2]
and [44 Section 32]) Fix a faithful G-module W Let
ϕ KGL(W ) rarr A (31)
be the differential epimorphism of differential Hopf algebras corresponding to the embedding
G rarr GL(W ) Set
H =G
which is a LAG Define
A0 =ϕ(K[GL(W )]) = K[H ] (32)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 13
and for n Ecirc 1
An = spanK
prodjisinJ
θ j y j isin A∣∣∣ J is a finite set y j isin A0 θ j isinΘ
sumjisinJ
ord(θ j ) Eacute n
(33)
The following shows that the subspaces An sub A form a filtration (in the sense of [55]) of the
Hopf algebra A
Proposition 31 We have
A = ⋃nisinN
An An sub An+1 (34)
Ai A j sub Ai+ j i j isinN (35)
∆(An) subnsum
i=0Ai otimesK Anminusi (36)
Proof Relation (35) follows immediately from (33) Since K[GL(W )] differentially generates
KGL(W ) and ϕ is a differential epimorphism A0 differentially generates A which implies (34)
Finally let us prove (36) Consider the differential Hopf algebra
B = AotimesK A
where partl 1 Eacute l Eacute m acts on B as follows
partl (x otimes y) = partl (x)otimes y +x otimespartl (y) x y isin A
Set
Bn =nsum
i=0Ai otimesK Anminusi n isinN
We have
Bi B j sub Bi+ j and partl (Bn) sub Bn+1 i j isinN n isinN 1 Eacute l Eacute m (37)
Since K[GL(W )] is a Hopf subalgebra of KGL(W ) A0 is a Hopf subalgebra of A In particular
∆(A0) sub B0 (38)
14 A Minchenko et al
Since ∆ A rarr B is a differential homomorphism definition (33) and relations (38) (37) imply
∆(An) sub Bn n isinN
We will call AnnisinN the W -filtration of A As the definition of An depends on W we will
sometimes write An(W ) for An By (36) An is a subcomodule of A If x isin A An then the relation
x = (εotimes Id)∆(x) (39)
shows that ∆(x) 6isin A otimes An Therefore An is the largest subcomodule U sub A such that ∆(U ) subU otimesK An This suggests the following notation
For V isin RepG and n isinN let Vn denote the largest submodule U subV such that
V (U ) subU otimesK An
Then submodules Vn subV n isinN form a filtration of V which we also call the W -filtration
Proposition 32 For a morphism f U rarrV of G-modules and an n isinN we have f (Un) subVn
Proof The proof follows immediately from the definition of a morphism of G-modules
Note that Un subVn and Vn capU subUn for all submodules U subV isin RepG Therefore
Un =U capVn for every subcomodule U subV isin RepG (310)
(U oplusV )n =Un oplusVn for all U V isin RepG (311)(⋃iisinNV (i )
)n =⋃
iisinNV (i )n V (i ) subV (i +1) isin RepG (312)
Proposition 33 For every V isin RepG we have
V (Vn) subnsum
i=0Vi otimesK Anminusi (313)
Proof Let X denote the set of all V isin RepG satisfying (313) It follows from (310) and (311) that
if U V isin X then every submodule of U oplusV belongs to X If V isin RepG then V is isomorphic to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 15
submodule of AdimV by Proposition 25 Since A isin X by Proposition 31 Ob(RepG) sub X For the
general case it remains to apply (312)
Recall that a module is called semisimple if it equals the sum of its simple submodules
Proposition 34 Suppose that W is a semisimple G-module Then the LAG H is reductive If W
is not semisimple then it is not semisimple as an H-module
Proof For the proof see [39 proof of Theorem 47]
Lemma 35 Let V isin RepG If V is semisimple then V = V0 (Loosely speaking this means that
all completely reducible representations of an LDAG are polynomial This was also proved in [39
Theorem 33]) If W is semisimple the converse is true
Proof By (311) it suffices to prove the statement for a simple V isin RepG Suppose that V is
simple and V =Vn 6=Vnminus1 Then Vnminus1 = 0 and Proposition 33 implies
V (V ) subV otimes A0 (314)
Hence V =V0
Suppose that W is semisimple and V = V0 isin RepG The latter means (314) that is the
representation of G on V extends to the representation of H on V But H is reductive by
Proposition 34 (since W is semisimple) Then V is semisimple as an H-module Again by
Proposition 34 the G-module V is semisimple
Corollary 36 If W is semisimple then A0 is the sum of all simple subcomodules of A Therefore
if U V are faithful semisimple G-modules then the U - and V -filtrations of A coincide
Proof By Lemma 35 if Z sub A is simple then Z = Z0 Hence by Proposition 32 Z is contained
in A0 Moreover by Lemma 35 A0 is the sum of all its simple submodules
Corollary 37 The LDAG G is connected if and only if the LAG H is connected
Proof If G is Kolchin connected and
A = KG = KGL(W )p= KXi j 1detp
16 A Minchenko et al
then the differential ideal p is prime [8 p 895] Since by [8 p 897]
A0 = K[H ] = K[GL(W )]
(pcapK[GL(W )]) = K[Xi j 1det]
(pcapK[Xi j 1det])
and the ideal pcapK[Xi j 1det] is prime H is Zariski connected
Set Γ =GG which is finite Denote the quotient map by
π G rarr Γ
Since Γ is finite and charK = 0 B = KΓ isin RepΓ is semisimple Then B has a structure of a
semisimple G-module via π Therefore by Lemma 35 B = B0 Since πlowast is a homomorphism of
G-modules by Proposition 32
πlowast(B) =πlowast(B0) sub A0 = K[H ]
This means that π is a restriction of an epimorphism H rarr Γ which completes the proof
For the ∆-field K denote the underlying abstract field endowed with the trivial differential
structure (partl k = 0 1 Eacute l Eacute m) by K
Proposition 38 Suppose that the LDAG G is connected If x isin Ai y isin A j and x y isin Ai+ jminus1 then
either x isin Aiminus1 or y isin A jminus1
Proof We need to show that the graded algebra
gr A = oplusnisinN
AnAnminus1
is an integral domain Note that gr A is a differential algebra via
partl (x + Anminus1) = partl (x)+ An x isin An
Furthermore to a homomorphism ν B rarrC of filtered algebras such that ν(Bn) subCn n isinN there
corresponds the homomorphism
grν grB rarr grC x +Bnminus1 7rarr ν(x)+Cnminus1 x isin Bn
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 17
Let us identify GL(W ) with GLd d = dimW and set
B =Qxi j 1det
the coordinate ring of GLd overQ The algebra B is graded by
B n = spanQ
prodjisinJ
θ j y j
∣∣∣ J is a finite set y j isinQ[GLd ] θ j isinΘsumjisinJ
ord(θ j ) = n
n isinN
The W -filtration of B is then associated with this grading
Bn =noplus
i=0B i
For a field extension Qsub L set LB = B otimesQ L a Hopf algebra over L Then the algebra LB is graded
by LB n = B n otimesL
Let I stand for the Hopf ideal of KB defining G sub GLd For x isin KB let xh denote the highest
degree component of x with respect to the grading
KB n Let I denote the K-span of all xh x isin I
As in the proof of Proposition 31 we conclude that for all n isinN
∆(B n
)sub nsumi=0
B i otimesK B nminusi (315)
Since ∆(I ) sub I otimesK B +B otimesK I inclusion (315) implies that for all n isinN and x isin I capBn
I otimesK Bn +Bn otimesK I 3∆(x) =∆(x minusxh)+∆(xh) isin(
nminus1sumi=0
Bi otimesK Bnminusiminus1
)oplus
(nsum
i=0B i otimesK B nminusi
)
Hence by induction one has
∆(xh) isin I otimesK Bn +Bn otimesK I sub I otimesK B +B otimesK I
We have S(I ) sub I where S B rarr B is the antipode Moreover since S(B0) = B0 and S is differential
S(B n
)sub B n n isinN
Hence
S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n
18 A Minchenko et al
which implies that
S(I)sub I
Therefore I is a Hopf ideal of KB (not necessarily differential) Consider the algebra map
α KBβ grK B
grϕminusrarr gr A
where β is defined by the sections
KB n rarr KBn
KBnminus1 n isinN
and ϕ is given by (31) For every x isin I let n isinN be such that xh isin B n Then
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 3
In this paper we also apply these results to the Galois theory of parameterized linear differ-
ential equations The classical differential Galois theory studies symmetry groups of solutions of
linear differential equations or equivalently the groups of automorphisms of the corresponding
extensions of differential fields The groups that arise are LAGs over the field of constants This
theory started in the 19th century by Picard and Vessiot was put on a firm modern footing by
Kolchin [32] A generalized differential Galois theory that uses Kolchinrsquos axiomatic approach [34]
and realizes differential algebraic groups as Galois groups was initiated in [36]
The PPV Galois theory considered by Cassidy and Singer in [14] is a special case of the
Landesman generalized differential Galois theory and studies symmetry groups of the solutions of
linear differential equations whose coefficients contain parameters This is done by constructing
a differential field containing the solutions and their derivatives with respect to the parameters
called a PPV extension and studying its group of differential symmetries called a parameterized
differential Galois group The Galois groups that arise are LDAGs which are defined by polynomial
differential equations in the parameters Another approach to the Galois theory of systems of
linear differential equations with parameters is given in [7] where the authors study Galois groups
for generic values of the parameters It was shown in [19 43] that a necessary and sufficient
condition that an LDAG G is a PPV-Galois group over the field C (x) is that G contains a finitely
generated Kolchin-dense subgroup (under some further restrictions on C )
In Section 5 we show how our main result yields algorithms in the PPV theory For systems
of differential equations without parameters in the usual PicardndashVessiot theory there are many
existing algorithms for computing differential Galois groups A complete algorithm over the field
C (x) where C is a computable algebraically closed field of constants x is transcendental over
C and its derivative is equal to 1 is given in [58] (see also [15] for the case when the group is
reductive) More efficient algorithms for equations of low order appear in [35 51 52 53 56 57]
These latter algorithms depend on knowing a list of groups that can possibly occur and step-by-
step eliminating the choices
For parameterized systems the first known algorithms are given in [1 18] which apply
to systems of first and second orders (see also [2] for the application of these techniques
to the incomplete gamma function) An algorithm for the case in which the quotient of the
parameterized Galois group by its unipotent radical is constant is given in [41] In the present
paper without any restrictions to the order of the equations based on our main result (upper
bounds mentioned above) we present algorithms that
1 compute the quotient of the parameterized Galois group G by its unipotent radical Ru(G)
2 test whether G is reductive (ie whether Ru(G) = id)
4 A Minchenko et al
Note that these algorithms imply that we can determine if the PPV-Galois group is reductive and
if it is compute it
The paper is organized as follows We start by recalling the basic definitions of differential
algebra differential dimension differential algebraic groups their representations and unipo-
tent and reductive differential algebraic groups in Section 2 The main technical tools of the paper
properties of LDAGs containing a Kolchin-dense finitely generated subgroup and grading filtra-
tions of differential coordinate rings can be found in Sections 223 and 3 respectively The main
result is in Section 4 The main algorithms are described in Section 5 Examples that show that the
main upper bound is sharp and illustrate the algorithm are in Section 6
2 Basic definitions
21 Differential algebra
We begin by fixing notation and recalling some basic facts from differential algebra (cf [33])
In this paper a ∆-ring will be a commutative associative ring R with unit 1 and commuting
derivations ∆= part1 partm We let
Θ = part
i11 middot middotpartim
m | i j Ecirc 0
and note that this free semigroup acts naturally on R For an element parti11 middot middotpartim
m isinΘ we let
ord(part
i11 middot middotpartim
m)
= i1 + + im
Let Y = y1 yn be a set of variables and
ΘY = θy j |θ isinΘ 1 Eacute j Eacute n
The ring of differential polynomials RY in differential indeterminates Y over R is R[ΘY ] with
the derivations parti that extends the parti -action on R as follows
parti(θy j
)= (parti middotθ)y j 1 Eacute j Eacute n 1 Eacute i Eacute m
An ideal I in a∆-ring R is called a differential ideal if parti (a) isin I for all a isin I 1 Eacute i Eacute m For F sub R [F ]
denotes the differential ideal of R generated by F
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 5
Let K be a ∆-field of characteristic zero We denote the subfield of constants of K by
K∆ = c isin K | parti (c) = 0 1 Eacute i Eacute m
Let U be a differentially closed field containing K that is a ∆- extension field of K such that any
system of polynomial differential equations with coefficients in U having a solution in some ∆-
extension of U already have a solution in U n (see [14 Definition 32] and the references therein)
Definition 21 A Kolchin-closed subset W (U ) of U n over K is the set of common zeroes of a
system of differential algebraic equations with coefficients in K that is for f1 fl isin KY we
define
W (U ) = a isinU n | f1(a) = = fl (a) = 0
If W (U ) is a Kolchin-closed subset of U n over K we let I(W ) = f isin Ky1 yn | f (w) = 0 forall w isinW (U )
One has the usual correspondence between Kolchin-closed subsets of Kn defined over K and
radical differential ideals of Ky1 yn Given a Kolchin-closed subset W of U n defined over K
we let the coordinate ring KW be defined as
KW = Ky1 ynI(W )
A differential polynomial map ϕ W1 rarr W2 between Kolchin-closed subsets of U n1 and U n2
respectively defined over K is given in coordinates by differential polynomials in KW1 More-
over to give ϕ W1 rarr W2 is equivalent to defining a differential K-homomorphism ϕlowast KW2 rarrKW1 If KW is an integral domain then W is called irreducible This is equivalent to I(W ) being
a prime differential ideal More generally if
I(W ) = p1 cap cappq
is a minimal prime decomposition which is unique up to permutation [30 VII29] then the
irreducible Kolchin-closed sets W1 Wq corresponding to p1 pq are called the irreducible
components of W We then have
W =W1 cup cupWq
If W is an irreducible Kolchin-closed subset of U n defined over K we denote the quotient field of
KW by KlangW rang
6 A Minchenko et al
In the following we shall need the notion of a Kolchin closed set being of differential type at
most zero The general concept of differential type is defined in terms of the Kolchin polynomial
([33 Section II12]) but this more restricted notion has a simpler definition
Definition 22 Let W be an irreducible Kolchin-closed subset of U n defined over K We say that
W is of differential type at most zero and denote this by τ(W ) Eacute 0 if tr degKKlangWrang ltinfin If W is an
arbitrary Kolchin-closed subset of U n defined over K we say that W has differential type at most
zero if this is true for each of its components
We shall use the fact that if H E G are LDAGs then τ(H) Eacute 0 and τ(GH) Eacute 0 if and only if
τ(G) Eacute 0 [34 Section IV4]
22 Linear Differential Algebraic Groups
Let K subU be as above Recall that LDAG stands for linear differential algebraic group
Definition 23 [8 Chapter II Section 1 p 905] An LDAG over K is a Kolchin-closed subgroup G
of GLn(U ) over K that is an intersection of a Kolchin-closed subset of U n2with GLn(U ) that is
closed under the group operations
Note that we identify GLn(U ) with a Zariski closed subset of U n2+1 given by
(A a) | (det(A)) middota minus1 = 0
If X is an invertible n timesn matrix we can identify it with the pair (X 1det(X )) Hence we may
represent the coordinate ring of GLn(U ) as KX 1det(X ) As usual let Gm(U ) and Ga(U ) denote
the multiplicative and additive groups of U respectively The coordinate ring of the LDAG SL2(U )
is isomorphic to
Kc11c12c21c22[c11c22 minus c12c21 minus1]
For a group G sub GLn(U ) we denote the Zariski closure of G in GLn(U ) by G Then G is a LAG over
U If G sub GLn(U ) is an LDAG defined over K then G is defined over K as well
The irreducible component of an LDAG G containing id the identity is called the identity
component of G and denoted by G An LDAG G is called connected if G =G which is equivalent
to G being an irreducible Kolchin closed set [8 p 906]
The coordinate ring KG of an LDAG G has a structure of a differential Hopf algebra that is
a Hopf algebra in which the coproduct antipode and counit are homomorphisms of differential
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 7
algebras [44 Section 32] and [9 Section 2] One can view G as a representable functor defined
on K-algebras represented by KG For example if V is an n-dimensional vector space over K
GL(V ) = AutV is an LDAG represented by KGLn = KGLn(U )
221 Representations of LDAGs
Definition 24 [9][44 Definition 6] Let G be an LDAG A differential polynomial group homo-
morphism
rV G rarr GL(V )
is called a differential representation of G where V is a finite-dimensional vector space over K
Such space is simply called a G-module This is equivalent to giving a comodule structure
ρV V rarrV otimesK KG
see [44 Definition 7 and Theorem 1] [59 Section 32] Moreover if U sub V is a submodule then
V |U = U
As usual morphisms between G-modules are K-linear maps that are G-equivariant The
category of differential representations of G is denoted by RepG
For an LDAG G let A = KG be its differential Hopf algebra and
∆ A rarr AotimesK A
be the comultiplication inducing the right-regular G-module structure on A as follows (see also
[44 Section 41]) For g x isinG(U ) and f isin A
(rg ( f )
)(x) = f (x middot g ) =∆( f )(x g ) =
nsumi=1
fi (x)gi (g )
where ∆( f ) =sumni=1 fi otimes gi The k-vector space A is an A-comodule via
A =∆
Proposition 25 [59 Corollary 33 Lemma 35][44 Lemma 3] The coalgebra A is a countable
union of its finite-dimensional subcoalgebras If V isin RepG then as an A-comodule V embeds
into AdimV
8 A Minchenko et al
By [8 Proposition 7] ρ(G) sub GL(V ) is a differential algebraic subgroup Given a representa-
tion ρ of an LDAG G one can define its prolongations
Pi (ρ) G rarr GL(Pi (V ))
with respect to parti as follows (see [21 Section 52] [44 Definition 4 and Theorem 1] and [39
p 1199]) Let
Pi (V ) = K ((KoplusKparti )K otimesK V ) (21)
as vector spaces where KoplusKparti is considered as the right K-module parti middota = parti (a)+aparti for all a isin K
Then the action of G is given by Pi (ρ) as follows
Pi (ρ)(g )(1otimes v) = 1otimesρ(g )(v) Pi (ρ)(g )(parti otimes v) = parti otimesρ(g )(v)
for all g isinG and v isinV In the language of matrices if Ag isin GLn corresponds to the action of g isinG
on V then the matrix (Ag parti Ag
0 Ag
)
corresponds to the action of g on Pi (V ) In what follows the q th iterate of Pi is denoted by P qi
Moreover the above induces the exact sequences
0 minusminusminusminusminusrarr Vιiminusminusminusminusminusrarr Pi (V )
πiminusminusminusminusminusrarr V minusminusminusminusminusrarr 0 (22)
where ιi (v) = 1otimesv and πi (aotimesu+bparti otimesv) = bv u v isinV a b isin K For any integer s we will refer to
P smP s
mminus1 middot middotP s1(ρ) G rarrGLNs
to be the s th total prolongation of ρ (where Ns is the dimension of the underlying prolonged
vector space) We denote this representation by P s (ρ) G rarr GLNs The underlying vector space
is denoted by P s (V )
It will be convenient to consider A as a G-module For this let RepG denote the differential
tensor category of all A-comodules (not necessarily finite-dimensional) which are direct limits of
finite-dimensional A-comodules by [59 Section 33] Then A isin RepG by Proposition 25
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 9
222 Unipotent radical of differential algebraic groups and reductive LDAGs
Definition 26 [10 Theorem 2] Let G be an LDAG defined over K We say that G is unipotent if
one of the following conditions holds
1 G is conjugate to a differential algebraic subgroup of the group Un of unipotent upper
triangular matrices
2 G contains no elements of finite order gt 1
3 G has a descending normal sequence of differential algebraic subgroups
G =G0 supG1 sup supGN = 1
with Gi Gi+1 isomorphic to a differential algebraic subgroup of the additive group Ga
One can show that an LDAG G defined over K admits a maximal normal unipotent differen-
tial subgroup [39 Theorem 310]
Definition 27 This subgroup is called the unipotent radical of G and denoted by Ru(G) The
unipotent radical of a LAG H is also denoted by Ru(H)
Definition 28 [39 Definition 312] An LDAG G is called reductive if its unipotent radical is trivial
that is Ru(G) = id
Remark 29 If G is given as a linear differential algebraic subgroup of some GLν we may consider
its Zariski closure G in GLν which is an algebraic group scheme defined over K Then following
the proof of [39 Theorem 310]
Ru(G) = Ru
(G
)capG
This implies that if G is reductive then G is reductive However in general the Zariski closure of
Ru(G) may be strictly included in Ru(G) [39 Ex 317]
223 Differentially finitely generated groups
As mentioned in the introduction one motivation for studying LDAGs is their use in the PPV
theory In Section 5 we will discuss PPV-extensions of certain fields whose PPV-Galois groups
satisfy the following property In this subsection we will assume that K is differentially closed
Definition 210 Let G be an LDAG defined over K We say that G is differentially finitely generated
or simply a DFGG if G(K) contains a finitely generated subgroup that is Kolchin dense over K
10 A Minchenko et al
Proposition 211 If G is a DFGG then its identity component G is a DFGG
Proof The ReidemeisterndashSchreier Theorem implies that a subgroup of finite index in a finitely
generated group is finitely generated ([38 Corollary 271]) One can use this fact to construct a
proof of the above Nonetheless we present a self-contained proof
Let F = GG and t = |GG| We claim that every sequence of t elements of F has a
contiguous subsequence whose product is the identity To see this let a1 at be a sequence
of elements of F Set
b1 = a1b2 = a1a2 bt = a1a2 middot middotat
If there are i lt j such that bi = b j then
id = bminus1i b j = a j+1 middot middota j
If the b j are pairwise distinct they exhaust F and so one of them must be the identity
Let S = Sminus1 be a finite set generating a dense subgroup ΓsubG Set
Γ0 = s | s = s1 middot middot sm isinG si isin S
Then Γ0 is a Kolchin dense subgroup of G Applying the above observation concerning F we see
that Γ0 is generated by the finite set
S0 = s | s = s1 middot middot sm isinG si isin S and m Eacute |GG|
Lemma 212 If H sub Gma is a DFGG then τ(H) Eacute 0
Proof Let πi be the projection of Gma onto its i th factor We have that πi (H) sub Ga is a DFGG and
so by [41 Lemma 210] τ(πi (H)) Eacute 0 Since
H subπ1(H)times timesπm(H) and τ(π1(H)times timesπm(H)) Eacute 0
we have τ(H) = 0
Lemma 213 If H sub Grm is a DFGG then τ(H) Eacute 0
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 11
Proof Let `∆ Grm rarr Gr m
a be the homomorphism
`∆(y1 yr ) =(part1 y1
y1
part1 yr
yrpart2 y1
y1
part2 yr
yr
partm y1
y1
partm yr
yr
)
The image of H under this homomorphism is a DFGG in Gr ma and so has differential type at most
0 The kernel of this homomorphism restricted to H is
(Gm
(K∆
))r capH
which also has type at most 0 Therefore τ(H) Eacute 0
Lemma 214 Let G be a reductive LDAG Then G is a DFGG if and only if τ(Z (G)
)Eacute 0
Proof Assume that G is a DFGG By Proposition 211 we can assume that G is Kolchin-connected
as well as a DFGG From [39 Theorem 47] we can assume that G = P is a reductive LAG From
the structure of reductive LAGs we know that
P = (PP ) middotZ (P )
where Z (P ) denotes the center (PP ) is the commutator subgroup and Z (P )cap (PP ) is finite Note
also that Z (P ) is a torus and that Z (G) = Z (P )capG Let
π P rarr P(PP ) Z (P )[Z (P )cap (PP )]
The image of G is connected and so lies in
π(Z (P )
) Gtm
for some t The image is a DFGG and so by Lemma 213 must have type at most 0 From the
description of π one sees that
π Z (G) rarr Z (G)[Z (P )cap (PP )] sub Z (P )[Z (P )cap (PP )]
Since Z (P )cap (PP ) is finite we have τ(Z (G)
)Eacute 0
Nowadays assume that τ(Z (G)
) Eacute 0 [41 Proposition 29] implies that Z (G) is a DFGG
Therefore it is enough to show that G prime = GZ (G) is a DFGG We see that G prime is semisimple and
12 A Minchenko et al
we will show that any semisimple LDAG is a DFGG Clearly it is enough to show that this is true
under the further assumption that G prime is connected
Let D be the K-vector space spanned by ∆ [13 Theorem 18] implies that G prime = G1 middot middotG`
where for each i there exists a simple LAG Hi defined over Q and a Lie (A Lie subspace E subD is
a subspace such that for any partpartprime isin E we have partpartprimeminuspartprimepart isin E ) K -subspace Ei of D such that
Gi = Hi
(KEi
) KEi =
c isin K | part(c) = 0 for all part isin Ei
Therefore it suffices to show that for a simple LAG H and a Lie K-subspace E sub D the LDAG
H(KE
)is a DFGG From [34 Proposition 6 and 7] E has a K-basis of commuting derivations Λ=
partprime1 partprimer which can be extended to a commuting basis
partprime1 partprimem
of D Let Π=
partprimer+1 partprimem
[14 Lemma 93] implies that KE is differentially closed as a Π-differential field We may consider
H(KE
)as a LAG over theΠ-differential field KE The result now follows from [50 Lemma 22]
3 Filtrations and gradings of the coordinate ring of an LDAG
In this section we develop the main technique of the paper filtrations and grading of coordinate
rings of LDAGs Let K be a ∆-field of characteristic zero not necessarily differentially closed The
set of natural numbers 012 is denoted byN
31 Filtrations of G-modules
Let G be an LDAG and A = KG be the corresponding differential Hopf algebra (see [9 Section 2]
and [44 Section 32]) Fix a faithful G-module W Let
ϕ KGL(W ) rarr A (31)
be the differential epimorphism of differential Hopf algebras corresponding to the embedding
G rarr GL(W ) Set
H =G
which is a LAG Define
A0 =ϕ(K[GL(W )]) = K[H ] (32)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 13
and for n Ecirc 1
An = spanK
prodjisinJ
θ j y j isin A∣∣∣ J is a finite set y j isin A0 θ j isinΘ
sumjisinJ
ord(θ j ) Eacute n
(33)
The following shows that the subspaces An sub A form a filtration (in the sense of [55]) of the
Hopf algebra A
Proposition 31 We have
A = ⋃nisinN
An An sub An+1 (34)
Ai A j sub Ai+ j i j isinN (35)
∆(An) subnsum
i=0Ai otimesK Anminusi (36)
Proof Relation (35) follows immediately from (33) Since K[GL(W )] differentially generates
KGL(W ) and ϕ is a differential epimorphism A0 differentially generates A which implies (34)
Finally let us prove (36) Consider the differential Hopf algebra
B = AotimesK A
where partl 1 Eacute l Eacute m acts on B as follows
partl (x otimes y) = partl (x)otimes y +x otimespartl (y) x y isin A
Set
Bn =nsum
i=0Ai otimesK Anminusi n isinN
We have
Bi B j sub Bi+ j and partl (Bn) sub Bn+1 i j isinN n isinN 1 Eacute l Eacute m (37)
Since K[GL(W )] is a Hopf subalgebra of KGL(W ) A0 is a Hopf subalgebra of A In particular
∆(A0) sub B0 (38)
14 A Minchenko et al
Since ∆ A rarr B is a differential homomorphism definition (33) and relations (38) (37) imply
∆(An) sub Bn n isinN
We will call AnnisinN the W -filtration of A As the definition of An depends on W we will
sometimes write An(W ) for An By (36) An is a subcomodule of A If x isin A An then the relation
x = (εotimes Id)∆(x) (39)
shows that ∆(x) 6isin A otimes An Therefore An is the largest subcomodule U sub A such that ∆(U ) subU otimesK An This suggests the following notation
For V isin RepG and n isinN let Vn denote the largest submodule U subV such that
V (U ) subU otimesK An
Then submodules Vn subV n isinN form a filtration of V which we also call the W -filtration
Proposition 32 For a morphism f U rarrV of G-modules and an n isinN we have f (Un) subVn
Proof The proof follows immediately from the definition of a morphism of G-modules
Note that Un subVn and Vn capU subUn for all submodules U subV isin RepG Therefore
Un =U capVn for every subcomodule U subV isin RepG (310)
(U oplusV )n =Un oplusVn for all U V isin RepG (311)(⋃iisinNV (i )
)n =⋃
iisinNV (i )n V (i ) subV (i +1) isin RepG (312)
Proposition 33 For every V isin RepG we have
V (Vn) subnsum
i=0Vi otimesK Anminusi (313)
Proof Let X denote the set of all V isin RepG satisfying (313) It follows from (310) and (311) that
if U V isin X then every submodule of U oplusV belongs to X If V isin RepG then V is isomorphic to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 15
submodule of AdimV by Proposition 25 Since A isin X by Proposition 31 Ob(RepG) sub X For the
general case it remains to apply (312)
Recall that a module is called semisimple if it equals the sum of its simple submodules
Proposition 34 Suppose that W is a semisimple G-module Then the LAG H is reductive If W
is not semisimple then it is not semisimple as an H-module
Proof For the proof see [39 proof of Theorem 47]
Lemma 35 Let V isin RepG If V is semisimple then V = V0 (Loosely speaking this means that
all completely reducible representations of an LDAG are polynomial This was also proved in [39
Theorem 33]) If W is semisimple the converse is true
Proof By (311) it suffices to prove the statement for a simple V isin RepG Suppose that V is
simple and V =Vn 6=Vnminus1 Then Vnminus1 = 0 and Proposition 33 implies
V (V ) subV otimes A0 (314)
Hence V =V0
Suppose that W is semisimple and V = V0 isin RepG The latter means (314) that is the
representation of G on V extends to the representation of H on V But H is reductive by
Proposition 34 (since W is semisimple) Then V is semisimple as an H-module Again by
Proposition 34 the G-module V is semisimple
Corollary 36 If W is semisimple then A0 is the sum of all simple subcomodules of A Therefore
if U V are faithful semisimple G-modules then the U - and V -filtrations of A coincide
Proof By Lemma 35 if Z sub A is simple then Z = Z0 Hence by Proposition 32 Z is contained
in A0 Moreover by Lemma 35 A0 is the sum of all its simple submodules
Corollary 37 The LDAG G is connected if and only if the LAG H is connected
Proof If G is Kolchin connected and
A = KG = KGL(W )p= KXi j 1detp
16 A Minchenko et al
then the differential ideal p is prime [8 p 895] Since by [8 p 897]
A0 = K[H ] = K[GL(W )]
(pcapK[GL(W )]) = K[Xi j 1det]
(pcapK[Xi j 1det])
and the ideal pcapK[Xi j 1det] is prime H is Zariski connected
Set Γ =GG which is finite Denote the quotient map by
π G rarr Γ
Since Γ is finite and charK = 0 B = KΓ isin RepΓ is semisimple Then B has a structure of a
semisimple G-module via π Therefore by Lemma 35 B = B0 Since πlowast is a homomorphism of
G-modules by Proposition 32
πlowast(B) =πlowast(B0) sub A0 = K[H ]
This means that π is a restriction of an epimorphism H rarr Γ which completes the proof
For the ∆-field K denote the underlying abstract field endowed with the trivial differential
structure (partl k = 0 1 Eacute l Eacute m) by K
Proposition 38 Suppose that the LDAG G is connected If x isin Ai y isin A j and x y isin Ai+ jminus1 then
either x isin Aiminus1 or y isin A jminus1
Proof We need to show that the graded algebra
gr A = oplusnisinN
AnAnminus1
is an integral domain Note that gr A is a differential algebra via
partl (x + Anminus1) = partl (x)+ An x isin An
Furthermore to a homomorphism ν B rarrC of filtered algebras such that ν(Bn) subCn n isinN there
corresponds the homomorphism
grν grB rarr grC x +Bnminus1 7rarr ν(x)+Cnminus1 x isin Bn
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 17
Let us identify GL(W ) with GLd d = dimW and set
B =Qxi j 1det
the coordinate ring of GLd overQ The algebra B is graded by
B n = spanQ
prodjisinJ
θ j y j
∣∣∣ J is a finite set y j isinQ[GLd ] θ j isinΘsumjisinJ
ord(θ j ) = n
n isinN
The W -filtration of B is then associated with this grading
Bn =noplus
i=0B i
For a field extension Qsub L set LB = B otimesQ L a Hopf algebra over L Then the algebra LB is graded
by LB n = B n otimesL
Let I stand for the Hopf ideal of KB defining G sub GLd For x isin KB let xh denote the highest
degree component of x with respect to the grading
KB n Let I denote the K-span of all xh x isin I
As in the proof of Proposition 31 we conclude that for all n isinN
∆(B n
)sub nsumi=0
B i otimesK B nminusi (315)
Since ∆(I ) sub I otimesK B +B otimesK I inclusion (315) implies that for all n isinN and x isin I capBn
I otimesK Bn +Bn otimesK I 3∆(x) =∆(x minusxh)+∆(xh) isin(
nminus1sumi=0
Bi otimesK Bnminusiminus1
)oplus
(nsum
i=0B i otimesK B nminusi
)
Hence by induction one has
∆(xh) isin I otimesK Bn +Bn otimesK I sub I otimesK B +B otimesK I
We have S(I ) sub I where S B rarr B is the antipode Moreover since S(B0) = B0 and S is differential
S(B n
)sub B n n isinN
Hence
S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n
18 A Minchenko et al
which implies that
S(I)sub I
Therefore I is a Hopf ideal of KB (not necessarily differential) Consider the algebra map
α KBβ grK B
grϕminusrarr gr A
where β is defined by the sections
KB n rarr KBn
KBnminus1 n isinN
and ϕ is given by (31) For every x isin I let n isinN be such that xh isin B n Then
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
4 A Minchenko et al
Note that these algorithms imply that we can determine if the PPV-Galois group is reductive and
if it is compute it
The paper is organized as follows We start by recalling the basic definitions of differential
algebra differential dimension differential algebraic groups their representations and unipo-
tent and reductive differential algebraic groups in Section 2 The main technical tools of the paper
properties of LDAGs containing a Kolchin-dense finitely generated subgroup and grading filtra-
tions of differential coordinate rings can be found in Sections 223 and 3 respectively The main
result is in Section 4 The main algorithms are described in Section 5 Examples that show that the
main upper bound is sharp and illustrate the algorithm are in Section 6
2 Basic definitions
21 Differential algebra
We begin by fixing notation and recalling some basic facts from differential algebra (cf [33])
In this paper a ∆-ring will be a commutative associative ring R with unit 1 and commuting
derivations ∆= part1 partm We let
Θ = part
i11 middot middotpartim
m | i j Ecirc 0
and note that this free semigroup acts naturally on R For an element parti11 middot middotpartim
m isinΘ we let
ord(part
i11 middot middotpartim
m)
= i1 + + im
Let Y = y1 yn be a set of variables and
ΘY = θy j |θ isinΘ 1 Eacute j Eacute n
The ring of differential polynomials RY in differential indeterminates Y over R is R[ΘY ] with
the derivations parti that extends the parti -action on R as follows
parti(θy j
)= (parti middotθ)y j 1 Eacute j Eacute n 1 Eacute i Eacute m
An ideal I in a∆-ring R is called a differential ideal if parti (a) isin I for all a isin I 1 Eacute i Eacute m For F sub R [F ]
denotes the differential ideal of R generated by F
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 5
Let K be a ∆-field of characteristic zero We denote the subfield of constants of K by
K∆ = c isin K | parti (c) = 0 1 Eacute i Eacute m
Let U be a differentially closed field containing K that is a ∆- extension field of K such that any
system of polynomial differential equations with coefficients in U having a solution in some ∆-
extension of U already have a solution in U n (see [14 Definition 32] and the references therein)
Definition 21 A Kolchin-closed subset W (U ) of U n over K is the set of common zeroes of a
system of differential algebraic equations with coefficients in K that is for f1 fl isin KY we
define
W (U ) = a isinU n | f1(a) = = fl (a) = 0
If W (U ) is a Kolchin-closed subset of U n over K we let I(W ) = f isin Ky1 yn | f (w) = 0 forall w isinW (U )
One has the usual correspondence between Kolchin-closed subsets of Kn defined over K and
radical differential ideals of Ky1 yn Given a Kolchin-closed subset W of U n defined over K
we let the coordinate ring KW be defined as
KW = Ky1 ynI(W )
A differential polynomial map ϕ W1 rarr W2 between Kolchin-closed subsets of U n1 and U n2
respectively defined over K is given in coordinates by differential polynomials in KW1 More-
over to give ϕ W1 rarr W2 is equivalent to defining a differential K-homomorphism ϕlowast KW2 rarrKW1 If KW is an integral domain then W is called irreducible This is equivalent to I(W ) being
a prime differential ideal More generally if
I(W ) = p1 cap cappq
is a minimal prime decomposition which is unique up to permutation [30 VII29] then the
irreducible Kolchin-closed sets W1 Wq corresponding to p1 pq are called the irreducible
components of W We then have
W =W1 cup cupWq
If W is an irreducible Kolchin-closed subset of U n defined over K we denote the quotient field of
KW by KlangW rang
6 A Minchenko et al
In the following we shall need the notion of a Kolchin closed set being of differential type at
most zero The general concept of differential type is defined in terms of the Kolchin polynomial
([33 Section II12]) but this more restricted notion has a simpler definition
Definition 22 Let W be an irreducible Kolchin-closed subset of U n defined over K We say that
W is of differential type at most zero and denote this by τ(W ) Eacute 0 if tr degKKlangWrang ltinfin If W is an
arbitrary Kolchin-closed subset of U n defined over K we say that W has differential type at most
zero if this is true for each of its components
We shall use the fact that if H E G are LDAGs then τ(H) Eacute 0 and τ(GH) Eacute 0 if and only if
τ(G) Eacute 0 [34 Section IV4]
22 Linear Differential Algebraic Groups
Let K subU be as above Recall that LDAG stands for linear differential algebraic group
Definition 23 [8 Chapter II Section 1 p 905] An LDAG over K is a Kolchin-closed subgroup G
of GLn(U ) over K that is an intersection of a Kolchin-closed subset of U n2with GLn(U ) that is
closed under the group operations
Note that we identify GLn(U ) with a Zariski closed subset of U n2+1 given by
(A a) | (det(A)) middota minus1 = 0
If X is an invertible n timesn matrix we can identify it with the pair (X 1det(X )) Hence we may
represent the coordinate ring of GLn(U ) as KX 1det(X ) As usual let Gm(U ) and Ga(U ) denote
the multiplicative and additive groups of U respectively The coordinate ring of the LDAG SL2(U )
is isomorphic to
Kc11c12c21c22[c11c22 minus c12c21 minus1]
For a group G sub GLn(U ) we denote the Zariski closure of G in GLn(U ) by G Then G is a LAG over
U If G sub GLn(U ) is an LDAG defined over K then G is defined over K as well
The irreducible component of an LDAG G containing id the identity is called the identity
component of G and denoted by G An LDAG G is called connected if G =G which is equivalent
to G being an irreducible Kolchin closed set [8 p 906]
The coordinate ring KG of an LDAG G has a structure of a differential Hopf algebra that is
a Hopf algebra in which the coproduct antipode and counit are homomorphisms of differential
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 7
algebras [44 Section 32] and [9 Section 2] One can view G as a representable functor defined
on K-algebras represented by KG For example if V is an n-dimensional vector space over K
GL(V ) = AutV is an LDAG represented by KGLn = KGLn(U )
221 Representations of LDAGs
Definition 24 [9][44 Definition 6] Let G be an LDAG A differential polynomial group homo-
morphism
rV G rarr GL(V )
is called a differential representation of G where V is a finite-dimensional vector space over K
Such space is simply called a G-module This is equivalent to giving a comodule structure
ρV V rarrV otimesK KG
see [44 Definition 7 and Theorem 1] [59 Section 32] Moreover if U sub V is a submodule then
V |U = U
As usual morphisms between G-modules are K-linear maps that are G-equivariant The
category of differential representations of G is denoted by RepG
For an LDAG G let A = KG be its differential Hopf algebra and
∆ A rarr AotimesK A
be the comultiplication inducing the right-regular G-module structure on A as follows (see also
[44 Section 41]) For g x isinG(U ) and f isin A
(rg ( f )
)(x) = f (x middot g ) =∆( f )(x g ) =
nsumi=1
fi (x)gi (g )
where ∆( f ) =sumni=1 fi otimes gi The k-vector space A is an A-comodule via
A =∆
Proposition 25 [59 Corollary 33 Lemma 35][44 Lemma 3] The coalgebra A is a countable
union of its finite-dimensional subcoalgebras If V isin RepG then as an A-comodule V embeds
into AdimV
8 A Minchenko et al
By [8 Proposition 7] ρ(G) sub GL(V ) is a differential algebraic subgroup Given a representa-
tion ρ of an LDAG G one can define its prolongations
Pi (ρ) G rarr GL(Pi (V ))
with respect to parti as follows (see [21 Section 52] [44 Definition 4 and Theorem 1] and [39
p 1199]) Let
Pi (V ) = K ((KoplusKparti )K otimesK V ) (21)
as vector spaces where KoplusKparti is considered as the right K-module parti middota = parti (a)+aparti for all a isin K
Then the action of G is given by Pi (ρ) as follows
Pi (ρ)(g )(1otimes v) = 1otimesρ(g )(v) Pi (ρ)(g )(parti otimes v) = parti otimesρ(g )(v)
for all g isinG and v isinV In the language of matrices if Ag isin GLn corresponds to the action of g isinG
on V then the matrix (Ag parti Ag
0 Ag
)
corresponds to the action of g on Pi (V ) In what follows the q th iterate of Pi is denoted by P qi
Moreover the above induces the exact sequences
0 minusminusminusminusminusrarr Vιiminusminusminusminusminusrarr Pi (V )
πiminusminusminusminusminusrarr V minusminusminusminusminusrarr 0 (22)
where ιi (v) = 1otimesv and πi (aotimesu+bparti otimesv) = bv u v isinV a b isin K For any integer s we will refer to
P smP s
mminus1 middot middotP s1(ρ) G rarrGLNs
to be the s th total prolongation of ρ (where Ns is the dimension of the underlying prolonged
vector space) We denote this representation by P s (ρ) G rarr GLNs The underlying vector space
is denoted by P s (V )
It will be convenient to consider A as a G-module For this let RepG denote the differential
tensor category of all A-comodules (not necessarily finite-dimensional) which are direct limits of
finite-dimensional A-comodules by [59 Section 33] Then A isin RepG by Proposition 25
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 9
222 Unipotent radical of differential algebraic groups and reductive LDAGs
Definition 26 [10 Theorem 2] Let G be an LDAG defined over K We say that G is unipotent if
one of the following conditions holds
1 G is conjugate to a differential algebraic subgroup of the group Un of unipotent upper
triangular matrices
2 G contains no elements of finite order gt 1
3 G has a descending normal sequence of differential algebraic subgroups
G =G0 supG1 sup supGN = 1
with Gi Gi+1 isomorphic to a differential algebraic subgroup of the additive group Ga
One can show that an LDAG G defined over K admits a maximal normal unipotent differen-
tial subgroup [39 Theorem 310]
Definition 27 This subgroup is called the unipotent radical of G and denoted by Ru(G) The
unipotent radical of a LAG H is also denoted by Ru(H)
Definition 28 [39 Definition 312] An LDAG G is called reductive if its unipotent radical is trivial
that is Ru(G) = id
Remark 29 If G is given as a linear differential algebraic subgroup of some GLν we may consider
its Zariski closure G in GLν which is an algebraic group scheme defined over K Then following
the proof of [39 Theorem 310]
Ru(G) = Ru
(G
)capG
This implies that if G is reductive then G is reductive However in general the Zariski closure of
Ru(G) may be strictly included in Ru(G) [39 Ex 317]
223 Differentially finitely generated groups
As mentioned in the introduction one motivation for studying LDAGs is their use in the PPV
theory In Section 5 we will discuss PPV-extensions of certain fields whose PPV-Galois groups
satisfy the following property In this subsection we will assume that K is differentially closed
Definition 210 Let G be an LDAG defined over K We say that G is differentially finitely generated
or simply a DFGG if G(K) contains a finitely generated subgroup that is Kolchin dense over K
10 A Minchenko et al
Proposition 211 If G is a DFGG then its identity component G is a DFGG
Proof The ReidemeisterndashSchreier Theorem implies that a subgroup of finite index in a finitely
generated group is finitely generated ([38 Corollary 271]) One can use this fact to construct a
proof of the above Nonetheless we present a self-contained proof
Let F = GG and t = |GG| We claim that every sequence of t elements of F has a
contiguous subsequence whose product is the identity To see this let a1 at be a sequence
of elements of F Set
b1 = a1b2 = a1a2 bt = a1a2 middot middotat
If there are i lt j such that bi = b j then
id = bminus1i b j = a j+1 middot middota j
If the b j are pairwise distinct they exhaust F and so one of them must be the identity
Let S = Sminus1 be a finite set generating a dense subgroup ΓsubG Set
Γ0 = s | s = s1 middot middot sm isinG si isin S
Then Γ0 is a Kolchin dense subgroup of G Applying the above observation concerning F we see
that Γ0 is generated by the finite set
S0 = s | s = s1 middot middot sm isinG si isin S and m Eacute |GG|
Lemma 212 If H sub Gma is a DFGG then τ(H) Eacute 0
Proof Let πi be the projection of Gma onto its i th factor We have that πi (H) sub Ga is a DFGG and
so by [41 Lemma 210] τ(πi (H)) Eacute 0 Since
H subπ1(H)times timesπm(H) and τ(π1(H)times timesπm(H)) Eacute 0
we have τ(H) = 0
Lemma 213 If H sub Grm is a DFGG then τ(H) Eacute 0
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 11
Proof Let `∆ Grm rarr Gr m
a be the homomorphism
`∆(y1 yr ) =(part1 y1
y1
part1 yr
yrpart2 y1
y1
part2 yr
yr
partm y1
y1
partm yr
yr
)
The image of H under this homomorphism is a DFGG in Gr ma and so has differential type at most
0 The kernel of this homomorphism restricted to H is
(Gm
(K∆
))r capH
which also has type at most 0 Therefore τ(H) Eacute 0
Lemma 214 Let G be a reductive LDAG Then G is a DFGG if and only if τ(Z (G)
)Eacute 0
Proof Assume that G is a DFGG By Proposition 211 we can assume that G is Kolchin-connected
as well as a DFGG From [39 Theorem 47] we can assume that G = P is a reductive LAG From
the structure of reductive LAGs we know that
P = (PP ) middotZ (P )
where Z (P ) denotes the center (PP ) is the commutator subgroup and Z (P )cap (PP ) is finite Note
also that Z (P ) is a torus and that Z (G) = Z (P )capG Let
π P rarr P(PP ) Z (P )[Z (P )cap (PP )]
The image of G is connected and so lies in
π(Z (P )
) Gtm
for some t The image is a DFGG and so by Lemma 213 must have type at most 0 From the
description of π one sees that
π Z (G) rarr Z (G)[Z (P )cap (PP )] sub Z (P )[Z (P )cap (PP )]
Since Z (P )cap (PP ) is finite we have τ(Z (G)
)Eacute 0
Nowadays assume that τ(Z (G)
) Eacute 0 [41 Proposition 29] implies that Z (G) is a DFGG
Therefore it is enough to show that G prime = GZ (G) is a DFGG We see that G prime is semisimple and
12 A Minchenko et al
we will show that any semisimple LDAG is a DFGG Clearly it is enough to show that this is true
under the further assumption that G prime is connected
Let D be the K-vector space spanned by ∆ [13 Theorem 18] implies that G prime = G1 middot middotG`
where for each i there exists a simple LAG Hi defined over Q and a Lie (A Lie subspace E subD is
a subspace such that for any partpartprime isin E we have partpartprimeminuspartprimepart isin E ) K -subspace Ei of D such that
Gi = Hi
(KEi
) KEi =
c isin K | part(c) = 0 for all part isin Ei
Therefore it suffices to show that for a simple LAG H and a Lie K-subspace E sub D the LDAG
H(KE
)is a DFGG From [34 Proposition 6 and 7] E has a K-basis of commuting derivations Λ=
partprime1 partprimer which can be extended to a commuting basis
partprime1 partprimem
of D Let Π=
partprimer+1 partprimem
[14 Lemma 93] implies that KE is differentially closed as a Π-differential field We may consider
H(KE
)as a LAG over theΠ-differential field KE The result now follows from [50 Lemma 22]
3 Filtrations and gradings of the coordinate ring of an LDAG
In this section we develop the main technique of the paper filtrations and grading of coordinate
rings of LDAGs Let K be a ∆-field of characteristic zero not necessarily differentially closed The
set of natural numbers 012 is denoted byN
31 Filtrations of G-modules
Let G be an LDAG and A = KG be the corresponding differential Hopf algebra (see [9 Section 2]
and [44 Section 32]) Fix a faithful G-module W Let
ϕ KGL(W ) rarr A (31)
be the differential epimorphism of differential Hopf algebras corresponding to the embedding
G rarr GL(W ) Set
H =G
which is a LAG Define
A0 =ϕ(K[GL(W )]) = K[H ] (32)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 13
and for n Ecirc 1
An = spanK
prodjisinJ
θ j y j isin A∣∣∣ J is a finite set y j isin A0 θ j isinΘ
sumjisinJ
ord(θ j ) Eacute n
(33)
The following shows that the subspaces An sub A form a filtration (in the sense of [55]) of the
Hopf algebra A
Proposition 31 We have
A = ⋃nisinN
An An sub An+1 (34)
Ai A j sub Ai+ j i j isinN (35)
∆(An) subnsum
i=0Ai otimesK Anminusi (36)
Proof Relation (35) follows immediately from (33) Since K[GL(W )] differentially generates
KGL(W ) and ϕ is a differential epimorphism A0 differentially generates A which implies (34)
Finally let us prove (36) Consider the differential Hopf algebra
B = AotimesK A
where partl 1 Eacute l Eacute m acts on B as follows
partl (x otimes y) = partl (x)otimes y +x otimespartl (y) x y isin A
Set
Bn =nsum
i=0Ai otimesK Anminusi n isinN
We have
Bi B j sub Bi+ j and partl (Bn) sub Bn+1 i j isinN n isinN 1 Eacute l Eacute m (37)
Since K[GL(W )] is a Hopf subalgebra of KGL(W ) A0 is a Hopf subalgebra of A In particular
∆(A0) sub B0 (38)
14 A Minchenko et al
Since ∆ A rarr B is a differential homomorphism definition (33) and relations (38) (37) imply
∆(An) sub Bn n isinN
We will call AnnisinN the W -filtration of A As the definition of An depends on W we will
sometimes write An(W ) for An By (36) An is a subcomodule of A If x isin A An then the relation
x = (εotimes Id)∆(x) (39)
shows that ∆(x) 6isin A otimes An Therefore An is the largest subcomodule U sub A such that ∆(U ) subU otimesK An This suggests the following notation
For V isin RepG and n isinN let Vn denote the largest submodule U subV such that
V (U ) subU otimesK An
Then submodules Vn subV n isinN form a filtration of V which we also call the W -filtration
Proposition 32 For a morphism f U rarrV of G-modules and an n isinN we have f (Un) subVn
Proof The proof follows immediately from the definition of a morphism of G-modules
Note that Un subVn and Vn capU subUn for all submodules U subV isin RepG Therefore
Un =U capVn for every subcomodule U subV isin RepG (310)
(U oplusV )n =Un oplusVn for all U V isin RepG (311)(⋃iisinNV (i )
)n =⋃
iisinNV (i )n V (i ) subV (i +1) isin RepG (312)
Proposition 33 For every V isin RepG we have
V (Vn) subnsum
i=0Vi otimesK Anminusi (313)
Proof Let X denote the set of all V isin RepG satisfying (313) It follows from (310) and (311) that
if U V isin X then every submodule of U oplusV belongs to X If V isin RepG then V is isomorphic to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 15
submodule of AdimV by Proposition 25 Since A isin X by Proposition 31 Ob(RepG) sub X For the
general case it remains to apply (312)
Recall that a module is called semisimple if it equals the sum of its simple submodules
Proposition 34 Suppose that W is a semisimple G-module Then the LAG H is reductive If W
is not semisimple then it is not semisimple as an H-module
Proof For the proof see [39 proof of Theorem 47]
Lemma 35 Let V isin RepG If V is semisimple then V = V0 (Loosely speaking this means that
all completely reducible representations of an LDAG are polynomial This was also proved in [39
Theorem 33]) If W is semisimple the converse is true
Proof By (311) it suffices to prove the statement for a simple V isin RepG Suppose that V is
simple and V =Vn 6=Vnminus1 Then Vnminus1 = 0 and Proposition 33 implies
V (V ) subV otimes A0 (314)
Hence V =V0
Suppose that W is semisimple and V = V0 isin RepG The latter means (314) that is the
representation of G on V extends to the representation of H on V But H is reductive by
Proposition 34 (since W is semisimple) Then V is semisimple as an H-module Again by
Proposition 34 the G-module V is semisimple
Corollary 36 If W is semisimple then A0 is the sum of all simple subcomodules of A Therefore
if U V are faithful semisimple G-modules then the U - and V -filtrations of A coincide
Proof By Lemma 35 if Z sub A is simple then Z = Z0 Hence by Proposition 32 Z is contained
in A0 Moreover by Lemma 35 A0 is the sum of all its simple submodules
Corollary 37 The LDAG G is connected if and only if the LAG H is connected
Proof If G is Kolchin connected and
A = KG = KGL(W )p= KXi j 1detp
16 A Minchenko et al
then the differential ideal p is prime [8 p 895] Since by [8 p 897]
A0 = K[H ] = K[GL(W )]
(pcapK[GL(W )]) = K[Xi j 1det]
(pcapK[Xi j 1det])
and the ideal pcapK[Xi j 1det] is prime H is Zariski connected
Set Γ =GG which is finite Denote the quotient map by
π G rarr Γ
Since Γ is finite and charK = 0 B = KΓ isin RepΓ is semisimple Then B has a structure of a
semisimple G-module via π Therefore by Lemma 35 B = B0 Since πlowast is a homomorphism of
G-modules by Proposition 32
πlowast(B) =πlowast(B0) sub A0 = K[H ]
This means that π is a restriction of an epimorphism H rarr Γ which completes the proof
For the ∆-field K denote the underlying abstract field endowed with the trivial differential
structure (partl k = 0 1 Eacute l Eacute m) by K
Proposition 38 Suppose that the LDAG G is connected If x isin Ai y isin A j and x y isin Ai+ jminus1 then
either x isin Aiminus1 or y isin A jminus1
Proof We need to show that the graded algebra
gr A = oplusnisinN
AnAnminus1
is an integral domain Note that gr A is a differential algebra via
partl (x + Anminus1) = partl (x)+ An x isin An
Furthermore to a homomorphism ν B rarrC of filtered algebras such that ν(Bn) subCn n isinN there
corresponds the homomorphism
grν grB rarr grC x +Bnminus1 7rarr ν(x)+Cnminus1 x isin Bn
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 17
Let us identify GL(W ) with GLd d = dimW and set
B =Qxi j 1det
the coordinate ring of GLd overQ The algebra B is graded by
B n = spanQ
prodjisinJ
θ j y j
∣∣∣ J is a finite set y j isinQ[GLd ] θ j isinΘsumjisinJ
ord(θ j ) = n
n isinN
The W -filtration of B is then associated with this grading
Bn =noplus
i=0B i
For a field extension Qsub L set LB = B otimesQ L a Hopf algebra over L Then the algebra LB is graded
by LB n = B n otimesL
Let I stand for the Hopf ideal of KB defining G sub GLd For x isin KB let xh denote the highest
degree component of x with respect to the grading
KB n Let I denote the K-span of all xh x isin I
As in the proof of Proposition 31 we conclude that for all n isinN
∆(B n
)sub nsumi=0
B i otimesK B nminusi (315)
Since ∆(I ) sub I otimesK B +B otimesK I inclusion (315) implies that for all n isinN and x isin I capBn
I otimesK Bn +Bn otimesK I 3∆(x) =∆(x minusxh)+∆(xh) isin(
nminus1sumi=0
Bi otimesK Bnminusiminus1
)oplus
(nsum
i=0B i otimesK B nminusi
)
Hence by induction one has
∆(xh) isin I otimesK Bn +Bn otimesK I sub I otimesK B +B otimesK I
We have S(I ) sub I where S B rarr B is the antipode Moreover since S(B0) = B0 and S is differential
S(B n
)sub B n n isinN
Hence
S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n
18 A Minchenko et al
which implies that
S(I)sub I
Therefore I is a Hopf ideal of KB (not necessarily differential) Consider the algebra map
α KBβ grK B
grϕminusrarr gr A
where β is defined by the sections
KB n rarr KBn
KBnminus1 n isinN
and ϕ is given by (31) For every x isin I let n isinN be such that xh isin B n Then
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 5
Let K be a ∆-field of characteristic zero We denote the subfield of constants of K by
K∆ = c isin K | parti (c) = 0 1 Eacute i Eacute m
Let U be a differentially closed field containing K that is a ∆- extension field of K such that any
system of polynomial differential equations with coefficients in U having a solution in some ∆-
extension of U already have a solution in U n (see [14 Definition 32] and the references therein)
Definition 21 A Kolchin-closed subset W (U ) of U n over K is the set of common zeroes of a
system of differential algebraic equations with coefficients in K that is for f1 fl isin KY we
define
W (U ) = a isinU n | f1(a) = = fl (a) = 0
If W (U ) is a Kolchin-closed subset of U n over K we let I(W ) = f isin Ky1 yn | f (w) = 0 forall w isinW (U )
One has the usual correspondence between Kolchin-closed subsets of Kn defined over K and
radical differential ideals of Ky1 yn Given a Kolchin-closed subset W of U n defined over K
we let the coordinate ring KW be defined as
KW = Ky1 ynI(W )
A differential polynomial map ϕ W1 rarr W2 between Kolchin-closed subsets of U n1 and U n2
respectively defined over K is given in coordinates by differential polynomials in KW1 More-
over to give ϕ W1 rarr W2 is equivalent to defining a differential K-homomorphism ϕlowast KW2 rarrKW1 If KW is an integral domain then W is called irreducible This is equivalent to I(W ) being
a prime differential ideal More generally if
I(W ) = p1 cap cappq
is a minimal prime decomposition which is unique up to permutation [30 VII29] then the
irreducible Kolchin-closed sets W1 Wq corresponding to p1 pq are called the irreducible
components of W We then have
W =W1 cup cupWq
If W is an irreducible Kolchin-closed subset of U n defined over K we denote the quotient field of
KW by KlangW rang
6 A Minchenko et al
In the following we shall need the notion of a Kolchin closed set being of differential type at
most zero The general concept of differential type is defined in terms of the Kolchin polynomial
([33 Section II12]) but this more restricted notion has a simpler definition
Definition 22 Let W be an irreducible Kolchin-closed subset of U n defined over K We say that
W is of differential type at most zero and denote this by τ(W ) Eacute 0 if tr degKKlangWrang ltinfin If W is an
arbitrary Kolchin-closed subset of U n defined over K we say that W has differential type at most
zero if this is true for each of its components
We shall use the fact that if H E G are LDAGs then τ(H) Eacute 0 and τ(GH) Eacute 0 if and only if
τ(G) Eacute 0 [34 Section IV4]
22 Linear Differential Algebraic Groups
Let K subU be as above Recall that LDAG stands for linear differential algebraic group
Definition 23 [8 Chapter II Section 1 p 905] An LDAG over K is a Kolchin-closed subgroup G
of GLn(U ) over K that is an intersection of a Kolchin-closed subset of U n2with GLn(U ) that is
closed under the group operations
Note that we identify GLn(U ) with a Zariski closed subset of U n2+1 given by
(A a) | (det(A)) middota minus1 = 0
If X is an invertible n timesn matrix we can identify it with the pair (X 1det(X )) Hence we may
represent the coordinate ring of GLn(U ) as KX 1det(X ) As usual let Gm(U ) and Ga(U ) denote
the multiplicative and additive groups of U respectively The coordinate ring of the LDAG SL2(U )
is isomorphic to
Kc11c12c21c22[c11c22 minus c12c21 minus1]
For a group G sub GLn(U ) we denote the Zariski closure of G in GLn(U ) by G Then G is a LAG over
U If G sub GLn(U ) is an LDAG defined over K then G is defined over K as well
The irreducible component of an LDAG G containing id the identity is called the identity
component of G and denoted by G An LDAG G is called connected if G =G which is equivalent
to G being an irreducible Kolchin closed set [8 p 906]
The coordinate ring KG of an LDAG G has a structure of a differential Hopf algebra that is
a Hopf algebra in which the coproduct antipode and counit are homomorphisms of differential
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 7
algebras [44 Section 32] and [9 Section 2] One can view G as a representable functor defined
on K-algebras represented by KG For example if V is an n-dimensional vector space over K
GL(V ) = AutV is an LDAG represented by KGLn = KGLn(U )
221 Representations of LDAGs
Definition 24 [9][44 Definition 6] Let G be an LDAG A differential polynomial group homo-
morphism
rV G rarr GL(V )
is called a differential representation of G where V is a finite-dimensional vector space over K
Such space is simply called a G-module This is equivalent to giving a comodule structure
ρV V rarrV otimesK KG
see [44 Definition 7 and Theorem 1] [59 Section 32] Moreover if U sub V is a submodule then
V |U = U
As usual morphisms between G-modules are K-linear maps that are G-equivariant The
category of differential representations of G is denoted by RepG
For an LDAG G let A = KG be its differential Hopf algebra and
∆ A rarr AotimesK A
be the comultiplication inducing the right-regular G-module structure on A as follows (see also
[44 Section 41]) For g x isinG(U ) and f isin A
(rg ( f )
)(x) = f (x middot g ) =∆( f )(x g ) =
nsumi=1
fi (x)gi (g )
where ∆( f ) =sumni=1 fi otimes gi The k-vector space A is an A-comodule via
A =∆
Proposition 25 [59 Corollary 33 Lemma 35][44 Lemma 3] The coalgebra A is a countable
union of its finite-dimensional subcoalgebras If V isin RepG then as an A-comodule V embeds
into AdimV
8 A Minchenko et al
By [8 Proposition 7] ρ(G) sub GL(V ) is a differential algebraic subgroup Given a representa-
tion ρ of an LDAG G one can define its prolongations
Pi (ρ) G rarr GL(Pi (V ))
with respect to parti as follows (see [21 Section 52] [44 Definition 4 and Theorem 1] and [39
p 1199]) Let
Pi (V ) = K ((KoplusKparti )K otimesK V ) (21)
as vector spaces where KoplusKparti is considered as the right K-module parti middota = parti (a)+aparti for all a isin K
Then the action of G is given by Pi (ρ) as follows
Pi (ρ)(g )(1otimes v) = 1otimesρ(g )(v) Pi (ρ)(g )(parti otimes v) = parti otimesρ(g )(v)
for all g isinG and v isinV In the language of matrices if Ag isin GLn corresponds to the action of g isinG
on V then the matrix (Ag parti Ag
0 Ag
)
corresponds to the action of g on Pi (V ) In what follows the q th iterate of Pi is denoted by P qi
Moreover the above induces the exact sequences
0 minusminusminusminusminusrarr Vιiminusminusminusminusminusrarr Pi (V )
πiminusminusminusminusminusrarr V minusminusminusminusminusrarr 0 (22)
where ιi (v) = 1otimesv and πi (aotimesu+bparti otimesv) = bv u v isinV a b isin K For any integer s we will refer to
P smP s
mminus1 middot middotP s1(ρ) G rarrGLNs
to be the s th total prolongation of ρ (where Ns is the dimension of the underlying prolonged
vector space) We denote this representation by P s (ρ) G rarr GLNs The underlying vector space
is denoted by P s (V )
It will be convenient to consider A as a G-module For this let RepG denote the differential
tensor category of all A-comodules (not necessarily finite-dimensional) which are direct limits of
finite-dimensional A-comodules by [59 Section 33] Then A isin RepG by Proposition 25
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 9
222 Unipotent radical of differential algebraic groups and reductive LDAGs
Definition 26 [10 Theorem 2] Let G be an LDAG defined over K We say that G is unipotent if
one of the following conditions holds
1 G is conjugate to a differential algebraic subgroup of the group Un of unipotent upper
triangular matrices
2 G contains no elements of finite order gt 1
3 G has a descending normal sequence of differential algebraic subgroups
G =G0 supG1 sup supGN = 1
with Gi Gi+1 isomorphic to a differential algebraic subgroup of the additive group Ga
One can show that an LDAG G defined over K admits a maximal normal unipotent differen-
tial subgroup [39 Theorem 310]
Definition 27 This subgroup is called the unipotent radical of G and denoted by Ru(G) The
unipotent radical of a LAG H is also denoted by Ru(H)
Definition 28 [39 Definition 312] An LDAG G is called reductive if its unipotent radical is trivial
that is Ru(G) = id
Remark 29 If G is given as a linear differential algebraic subgroup of some GLν we may consider
its Zariski closure G in GLν which is an algebraic group scheme defined over K Then following
the proof of [39 Theorem 310]
Ru(G) = Ru
(G
)capG
This implies that if G is reductive then G is reductive However in general the Zariski closure of
Ru(G) may be strictly included in Ru(G) [39 Ex 317]
223 Differentially finitely generated groups
As mentioned in the introduction one motivation for studying LDAGs is their use in the PPV
theory In Section 5 we will discuss PPV-extensions of certain fields whose PPV-Galois groups
satisfy the following property In this subsection we will assume that K is differentially closed
Definition 210 Let G be an LDAG defined over K We say that G is differentially finitely generated
or simply a DFGG if G(K) contains a finitely generated subgroup that is Kolchin dense over K
10 A Minchenko et al
Proposition 211 If G is a DFGG then its identity component G is a DFGG
Proof The ReidemeisterndashSchreier Theorem implies that a subgroup of finite index in a finitely
generated group is finitely generated ([38 Corollary 271]) One can use this fact to construct a
proof of the above Nonetheless we present a self-contained proof
Let F = GG and t = |GG| We claim that every sequence of t elements of F has a
contiguous subsequence whose product is the identity To see this let a1 at be a sequence
of elements of F Set
b1 = a1b2 = a1a2 bt = a1a2 middot middotat
If there are i lt j such that bi = b j then
id = bminus1i b j = a j+1 middot middota j
If the b j are pairwise distinct they exhaust F and so one of them must be the identity
Let S = Sminus1 be a finite set generating a dense subgroup ΓsubG Set
Γ0 = s | s = s1 middot middot sm isinG si isin S
Then Γ0 is a Kolchin dense subgroup of G Applying the above observation concerning F we see
that Γ0 is generated by the finite set
S0 = s | s = s1 middot middot sm isinG si isin S and m Eacute |GG|
Lemma 212 If H sub Gma is a DFGG then τ(H) Eacute 0
Proof Let πi be the projection of Gma onto its i th factor We have that πi (H) sub Ga is a DFGG and
so by [41 Lemma 210] τ(πi (H)) Eacute 0 Since
H subπ1(H)times timesπm(H) and τ(π1(H)times timesπm(H)) Eacute 0
we have τ(H) = 0
Lemma 213 If H sub Grm is a DFGG then τ(H) Eacute 0
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 11
Proof Let `∆ Grm rarr Gr m
a be the homomorphism
`∆(y1 yr ) =(part1 y1
y1
part1 yr
yrpart2 y1
y1
part2 yr
yr
partm y1
y1
partm yr
yr
)
The image of H under this homomorphism is a DFGG in Gr ma and so has differential type at most
0 The kernel of this homomorphism restricted to H is
(Gm
(K∆
))r capH
which also has type at most 0 Therefore τ(H) Eacute 0
Lemma 214 Let G be a reductive LDAG Then G is a DFGG if and only if τ(Z (G)
)Eacute 0
Proof Assume that G is a DFGG By Proposition 211 we can assume that G is Kolchin-connected
as well as a DFGG From [39 Theorem 47] we can assume that G = P is a reductive LAG From
the structure of reductive LAGs we know that
P = (PP ) middotZ (P )
where Z (P ) denotes the center (PP ) is the commutator subgroup and Z (P )cap (PP ) is finite Note
also that Z (P ) is a torus and that Z (G) = Z (P )capG Let
π P rarr P(PP ) Z (P )[Z (P )cap (PP )]
The image of G is connected and so lies in
π(Z (P )
) Gtm
for some t The image is a DFGG and so by Lemma 213 must have type at most 0 From the
description of π one sees that
π Z (G) rarr Z (G)[Z (P )cap (PP )] sub Z (P )[Z (P )cap (PP )]
Since Z (P )cap (PP ) is finite we have τ(Z (G)
)Eacute 0
Nowadays assume that τ(Z (G)
) Eacute 0 [41 Proposition 29] implies that Z (G) is a DFGG
Therefore it is enough to show that G prime = GZ (G) is a DFGG We see that G prime is semisimple and
12 A Minchenko et al
we will show that any semisimple LDAG is a DFGG Clearly it is enough to show that this is true
under the further assumption that G prime is connected
Let D be the K-vector space spanned by ∆ [13 Theorem 18] implies that G prime = G1 middot middotG`
where for each i there exists a simple LAG Hi defined over Q and a Lie (A Lie subspace E subD is
a subspace such that for any partpartprime isin E we have partpartprimeminuspartprimepart isin E ) K -subspace Ei of D such that
Gi = Hi
(KEi
) KEi =
c isin K | part(c) = 0 for all part isin Ei
Therefore it suffices to show that for a simple LAG H and a Lie K-subspace E sub D the LDAG
H(KE
)is a DFGG From [34 Proposition 6 and 7] E has a K-basis of commuting derivations Λ=
partprime1 partprimer which can be extended to a commuting basis
partprime1 partprimem
of D Let Π=
partprimer+1 partprimem
[14 Lemma 93] implies that KE is differentially closed as a Π-differential field We may consider
H(KE
)as a LAG over theΠ-differential field KE The result now follows from [50 Lemma 22]
3 Filtrations and gradings of the coordinate ring of an LDAG
In this section we develop the main technique of the paper filtrations and grading of coordinate
rings of LDAGs Let K be a ∆-field of characteristic zero not necessarily differentially closed The
set of natural numbers 012 is denoted byN
31 Filtrations of G-modules
Let G be an LDAG and A = KG be the corresponding differential Hopf algebra (see [9 Section 2]
and [44 Section 32]) Fix a faithful G-module W Let
ϕ KGL(W ) rarr A (31)
be the differential epimorphism of differential Hopf algebras corresponding to the embedding
G rarr GL(W ) Set
H =G
which is a LAG Define
A0 =ϕ(K[GL(W )]) = K[H ] (32)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 13
and for n Ecirc 1
An = spanK
prodjisinJ
θ j y j isin A∣∣∣ J is a finite set y j isin A0 θ j isinΘ
sumjisinJ
ord(θ j ) Eacute n
(33)
The following shows that the subspaces An sub A form a filtration (in the sense of [55]) of the
Hopf algebra A
Proposition 31 We have
A = ⋃nisinN
An An sub An+1 (34)
Ai A j sub Ai+ j i j isinN (35)
∆(An) subnsum
i=0Ai otimesK Anminusi (36)
Proof Relation (35) follows immediately from (33) Since K[GL(W )] differentially generates
KGL(W ) and ϕ is a differential epimorphism A0 differentially generates A which implies (34)
Finally let us prove (36) Consider the differential Hopf algebra
B = AotimesK A
where partl 1 Eacute l Eacute m acts on B as follows
partl (x otimes y) = partl (x)otimes y +x otimespartl (y) x y isin A
Set
Bn =nsum
i=0Ai otimesK Anminusi n isinN
We have
Bi B j sub Bi+ j and partl (Bn) sub Bn+1 i j isinN n isinN 1 Eacute l Eacute m (37)
Since K[GL(W )] is a Hopf subalgebra of KGL(W ) A0 is a Hopf subalgebra of A In particular
∆(A0) sub B0 (38)
14 A Minchenko et al
Since ∆ A rarr B is a differential homomorphism definition (33) and relations (38) (37) imply
∆(An) sub Bn n isinN
We will call AnnisinN the W -filtration of A As the definition of An depends on W we will
sometimes write An(W ) for An By (36) An is a subcomodule of A If x isin A An then the relation
x = (εotimes Id)∆(x) (39)
shows that ∆(x) 6isin A otimes An Therefore An is the largest subcomodule U sub A such that ∆(U ) subU otimesK An This suggests the following notation
For V isin RepG and n isinN let Vn denote the largest submodule U subV such that
V (U ) subU otimesK An
Then submodules Vn subV n isinN form a filtration of V which we also call the W -filtration
Proposition 32 For a morphism f U rarrV of G-modules and an n isinN we have f (Un) subVn
Proof The proof follows immediately from the definition of a morphism of G-modules
Note that Un subVn and Vn capU subUn for all submodules U subV isin RepG Therefore
Un =U capVn for every subcomodule U subV isin RepG (310)
(U oplusV )n =Un oplusVn for all U V isin RepG (311)(⋃iisinNV (i )
)n =⋃
iisinNV (i )n V (i ) subV (i +1) isin RepG (312)
Proposition 33 For every V isin RepG we have
V (Vn) subnsum
i=0Vi otimesK Anminusi (313)
Proof Let X denote the set of all V isin RepG satisfying (313) It follows from (310) and (311) that
if U V isin X then every submodule of U oplusV belongs to X If V isin RepG then V is isomorphic to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 15
submodule of AdimV by Proposition 25 Since A isin X by Proposition 31 Ob(RepG) sub X For the
general case it remains to apply (312)
Recall that a module is called semisimple if it equals the sum of its simple submodules
Proposition 34 Suppose that W is a semisimple G-module Then the LAG H is reductive If W
is not semisimple then it is not semisimple as an H-module
Proof For the proof see [39 proof of Theorem 47]
Lemma 35 Let V isin RepG If V is semisimple then V = V0 (Loosely speaking this means that
all completely reducible representations of an LDAG are polynomial This was also proved in [39
Theorem 33]) If W is semisimple the converse is true
Proof By (311) it suffices to prove the statement for a simple V isin RepG Suppose that V is
simple and V =Vn 6=Vnminus1 Then Vnminus1 = 0 and Proposition 33 implies
V (V ) subV otimes A0 (314)
Hence V =V0
Suppose that W is semisimple and V = V0 isin RepG The latter means (314) that is the
representation of G on V extends to the representation of H on V But H is reductive by
Proposition 34 (since W is semisimple) Then V is semisimple as an H-module Again by
Proposition 34 the G-module V is semisimple
Corollary 36 If W is semisimple then A0 is the sum of all simple subcomodules of A Therefore
if U V are faithful semisimple G-modules then the U - and V -filtrations of A coincide
Proof By Lemma 35 if Z sub A is simple then Z = Z0 Hence by Proposition 32 Z is contained
in A0 Moreover by Lemma 35 A0 is the sum of all its simple submodules
Corollary 37 The LDAG G is connected if and only if the LAG H is connected
Proof If G is Kolchin connected and
A = KG = KGL(W )p= KXi j 1detp
16 A Minchenko et al
then the differential ideal p is prime [8 p 895] Since by [8 p 897]
A0 = K[H ] = K[GL(W )]
(pcapK[GL(W )]) = K[Xi j 1det]
(pcapK[Xi j 1det])
and the ideal pcapK[Xi j 1det] is prime H is Zariski connected
Set Γ =GG which is finite Denote the quotient map by
π G rarr Γ
Since Γ is finite and charK = 0 B = KΓ isin RepΓ is semisimple Then B has a structure of a
semisimple G-module via π Therefore by Lemma 35 B = B0 Since πlowast is a homomorphism of
G-modules by Proposition 32
πlowast(B) =πlowast(B0) sub A0 = K[H ]
This means that π is a restriction of an epimorphism H rarr Γ which completes the proof
For the ∆-field K denote the underlying abstract field endowed with the trivial differential
structure (partl k = 0 1 Eacute l Eacute m) by K
Proposition 38 Suppose that the LDAG G is connected If x isin Ai y isin A j and x y isin Ai+ jminus1 then
either x isin Aiminus1 or y isin A jminus1
Proof We need to show that the graded algebra
gr A = oplusnisinN
AnAnminus1
is an integral domain Note that gr A is a differential algebra via
partl (x + Anminus1) = partl (x)+ An x isin An
Furthermore to a homomorphism ν B rarrC of filtered algebras such that ν(Bn) subCn n isinN there
corresponds the homomorphism
grν grB rarr grC x +Bnminus1 7rarr ν(x)+Cnminus1 x isin Bn
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 17
Let us identify GL(W ) with GLd d = dimW and set
B =Qxi j 1det
the coordinate ring of GLd overQ The algebra B is graded by
B n = spanQ
prodjisinJ
θ j y j
∣∣∣ J is a finite set y j isinQ[GLd ] θ j isinΘsumjisinJ
ord(θ j ) = n
n isinN
The W -filtration of B is then associated with this grading
Bn =noplus
i=0B i
For a field extension Qsub L set LB = B otimesQ L a Hopf algebra over L Then the algebra LB is graded
by LB n = B n otimesL
Let I stand for the Hopf ideal of KB defining G sub GLd For x isin KB let xh denote the highest
degree component of x with respect to the grading
KB n Let I denote the K-span of all xh x isin I
As in the proof of Proposition 31 we conclude that for all n isinN
∆(B n
)sub nsumi=0
B i otimesK B nminusi (315)
Since ∆(I ) sub I otimesK B +B otimesK I inclusion (315) implies that for all n isinN and x isin I capBn
I otimesK Bn +Bn otimesK I 3∆(x) =∆(x minusxh)+∆(xh) isin(
nminus1sumi=0
Bi otimesK Bnminusiminus1
)oplus
(nsum
i=0B i otimesK B nminusi
)
Hence by induction one has
∆(xh) isin I otimesK Bn +Bn otimesK I sub I otimesK B +B otimesK I
We have S(I ) sub I where S B rarr B is the antipode Moreover since S(B0) = B0 and S is differential
S(B n
)sub B n n isinN
Hence
S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n
18 A Minchenko et al
which implies that
S(I)sub I
Therefore I is a Hopf ideal of KB (not necessarily differential) Consider the algebra map
α KBβ grK B
grϕminusrarr gr A
where β is defined by the sections
KB n rarr KBn
KBnminus1 n isinN
and ϕ is given by (31) For every x isin I let n isinN be such that xh isin B n Then
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
Theorem 310 Let xi isin A 1 Eacute i Eacute r and x = x1x2 middot middotxr isin Ast Then for all i 1 Eacute i Eacute r there exist
si ti isinN such that xi isin Asi ti and
sumi
si Eacute s and maxi
ti Eacute t
Proof It suffices to consider only the case r = 2 Then Propositions 38 and 39 complete the
proof
For V isin RepG and n isin N let V(n) denote the largest submodule U of V such that V (U ) subU otimes A(n) (If V = A then V(n) = A(n) which follows from (39)) Similarly we define Vst s t isinN
For a reductive LDAG G and its coordinate ring A = KG let AnnisinN denote the W -filtration
corresponding to an arbitrary faithful semisimple G-module W This filtration does not depend
on the choice of W by Corollary 36
Definition 311 If φ G rarr L is a homomorphism of LDAGs and V isin RepL then φ induces the
structure of a G-module on V This G-module will be denoted by GV
Proposition 312 Let φ G rarr L be a homomorphism of reductive LDAGs Then
φlowast(Bst
)sub Ast s t isinN (318)
where A = KG and B = KL Suppose that Kerφ is finite and the index of φ(G) in L is finite
Then for every V isin RepL
V =Vst lArrrArr GV = (GV )st s t isinN (319)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 21
Proof Applying Lemma 35 to V = B0 and Proposition 32 to φlowast we obtain φlowast(B0) sub A0 Since
φlowast is a differential homomorphism relation (318) follows
Let us prove the second statement of the Proposition Note that the implication rArr of (319)
follows directly from (318) We will prove the implication lArr It suffices to consider two cases
1 G is connected and φ is injective
2 G is connected and φ is surjective
which follows from the commutative diagram
G φ|Gminusminusminusminusminusrarr Ly yG
φminusminusminusminusminusrarr L
Moreover by (312) and Proposition 25 it suffices to consider the case of finite-dimensional V
By the same proposition there is an embedding of L-modules
η V rarr B d d = dimV
Then GV is isomorphic to φlowastdη(V ) where φlowast
d B d rarr Ad is the application of φlowast componentwise
If GV = (GV )st then φlowastdη(V ) sub Ad
st Hence setting V (i ) to be the projection of η(V ) to the i th
component of B d we conclude φlowast(V (i )) sub Ast for all i 1 Eacute i Eacute d If we show that this implies
V (i ) sub Bst we are done So we will show that if V sub B then
φlowast(V ) =φlowast(V )st =rArrV =Vst
Case (i) Let us identify G with L via φ Suppose L sub GL(U ) where U is a semisimple L-module
Let g1 = 1 gr isin L be representatives of the cosets of L Let I ( j ) sub B 1 Eacute j Eacute r be the differential
ideal of functions vanishing on all connected components of L but g j L We have
B =roplus
j=1I ( j ) and I ( j ) = g j I (1)
The G-modules I = I (1) and A are isomorphic and the projection B rarr I corresponds to the
restriction mapφlowast The G-module structure on I ( j ) is obtained by the twist by conjugation G rarrG
22 A Minchenko et al
g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude
g j (In) = (g j I
)n
By Corollary 37 Zariski closures of connected components of L sub GL(U ) are connected compo-
nents of L Therefore
B0 =roplus
j=1g j (I0)
Then B0 cap I = I0 Since I is a differential ideal Bn cap I = In for all n isinN Let
v isinVn Vnminus1 (320)
Then for each j 1 Eacute i Eacute r there exists v( j ) isin I ( j ) such that
v =rsum
j=1v( j )
By (320) there exists j 1 Eacute j Eacute r such that v( j ) isinVn Vnminus1 Set
w = gminus1j v isinVn Vnminus1
Then by the above
φlowast(w) isin An Anminus1
We conclude that for all n isinN
φlowast(V ) =φlowast(V )n =rArr V =Vn
Similarly one can show that
φlowast(V ) =φlowast(V )(n) =rArr V =V(n)
Since Vst =Vs capV(t ) this completes the proof of Case (i)
Case (ii) Consider B as a subalgebra of A via φlowast It suffices to show
Ast capB sub Bst (321)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 23
We have B sub AΓ where Γ = Kerφ
Let us show that B0 = AΓ0 For this consider G and L as differential algebraic Zariski dense
subgroups of reductive LAGs Since B0 sub A0 the map φ extends to an epimorphism
φ G rarr L
Since Γ= Γ Γ is normal in G Hence φ factors through the epimorphism
micro GΓrarr L
If K is the image of G in the quotient GΓ then micro(K ) = L and micro is an isomorphism on K This
means that microlowast extends to an isomorphism of B = KL onto KK Since K is reductive the
isomorphism preserves the grading by the first part of the proposition In particular microlowast(B0) =KK 0 As K is dense in GΓ we obtain
B0 = K[L]= K
[GΓ
]= K[G
]Γ = AΓ0
Let us consider the following sets
Ast = x isin (Ast )Γ | exist0 6= b isin B0 bx isin Bst
s t isinN
These are B0-submodules of A (via multiplication) satisfying (317) as one can check Moreover
for every l 1 Eacute l Eacute m
partl(
Ast)sub As+1t+1 (322)
Indeed let x isin Ast b isin B0 and bx isin Bst Then
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
It is convenient sometimes to consider the Zariski closure H of G sub GL(W ) as an LDAG
To distinguish the structures let us denote the latter by H diff Then Rep H diff is identified with a
subcategory of RepG
Lemma 43 If H is reductive then (44) holds for all U V isin Rep H diff and n isinN
Proof By Proposition 42 we only need to prove the formula for n = 1 Since A20 = A0 we have
by Lemma 35
(socU )otimes (socV ) =U0 otimesV0 sub (U otimesV )0 = soc(U otimesV )
Let us prove the other inclusion Since charK = 0
soc(U otimesK L) = (socU )otimesK L
for all differential field extensions L sup K by [6 Section 7] Therefore without loss of generality we
will assume that K is algebraically closed Moreover by Lemma 35 and Proposition 312 an H diff-
module is semisimple if and only if it is semisimple as an(H diff
)-module Therefore it suffices to
consider only the case of connected H Since a connected reductive group over an algebraically
closed field is defined overQ and the defining equations of H diff are of order 0 the W -filtration of
B = K
H diff
is associated with a grading (see proof of Proposition 38) In particular the sum I of
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
30 A Minchenko et al
all grading components but B0 = K[H ] is an ideal of B We have
B = B0 oplus I
Since B is an integral domain it follows that if x y isin B and x y isin B0 then x y isin B0 Hence
(U otimesV )0 subU0 otimesV0
which completes the proof
Proposition 44 For all V isin RepG
Vn sub socn+1 V
Proof We will use induction on n isin N with the case n = 0 being done by Lemma 35 Suppose
n Ecirc 1 and
Vnminus1 sub socn V
We need to show that the G-module
W = (Vn + socn V
)socn V Vn
(Vn cap socn V
)is semisimple But the latter is isomorphic to a quotient of U =VnVnminus1 since
Vnminus1 subVn cap socn V
By Proposition 33 U =U0 Finally Lemma 35 implies that U hence W is semisimple
42 Main result for semisimple LDAGs
Theorem 45 If G is semisimple then for all V isin RepG and n isinN
Vn = socn+1 V
Proof By Proposition 44 it suffices to prove that for all V isin RepG and n isinN
socn+1 V subVn (45)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 31
Let X sub Ob(RepG) denote the family of all V satisfying (45) for all n isinN We have by Lemma 35
V isin X for all semisimple V Suppose that V W isin Rep H diff sub RepG belong to X Then V oplusW and
V otimesW belong to X Indeed by Propositions 33 and 41 and Lemma 43
socn+1(V oplusW ) = socn+1 V oplus socn+1 W subVn oplusWn = (V oplusW )n
and
socn+1(V otimesW ) =nsum
i=0
(soci+1 V
)otimes (socn+1minusi W
)sub nsumi=0
Vi otimesWnminusi sub (V otimesW )n
Similarly Proposition 41 and (310) imply that if V isin X then all possible submodules and dif-
ferential prolongations of V belong to X Since RepG is differentially generated by a semisimple
V isin Rep H it remains only to check the following
If V isin RepG satisfies (45) then so do the dual V or and a quotient V U where U isin RepG
Since G is semisimple [13 Theorem 18] implies that G(U ) U a differentially closed field
containing K is differentially isomorphic to a group of the form G1 middotG2 middot middotGt where for each
i there is an algebraically closed field U i such that Gi is differentially isomorphic to the U i
points of a simple algebraic group Hi Since Hi = [Hi Hi ] we have G = [GG] and so we
must have G sub SL(V ) The group SL(V ) acts on V otimesdimV and has a nontrivial invariant element
corresponding to the determinant We conclude that for
r = |GG|dimV
the SL(V )-module V otimesr has a nontrivial G-invariant element Let E sub GL(V ) be the group
generated by SL(V ) and G Then the space
HomE(V orV otimesrminus1) (
V otimesr )E (46)
is nontrivial Since V or is a simple E-module this means that there exists an embedding
V or rarrV otimesrminus1
of E-modules and hence of G-modules Then V or isin X Finally since (V U )or embeds into V or it
belongs to X Then its dual V U isin X Hence X = Ob(RepG)
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
32 A Minchenko et al
43 Reductive case
Proposition 46 Let S and T be reductive LDAGs and G = S timesT For V isin RepG if SV = (SV )s1t1
and T V = (T V )s2t2 then V =Vs1+s2maxt1t2 (see Definition 311)
Proof We need to show that V = Vs1+s2 and V = V(maxt1t2) By Proposition 25 V embeds into
the G-module
U =dimVoplus
i=1A(i )
where A(i ) = A = B otimesK C where B = KS and C = KT We will identify V with its image in U
Let B j j isinN be subspaces of B such that
B j = B jminus1 oplus B j
Similarly we define subspaces Cr subC r isinN We have
A =oplusj r
B j otimesK Cr
as vector spaces Let
πij r U rarr A(i ) = A rarr B j otimesK Cr
denote the composition of the projections Then the conditions SV = (SV )s1 and SV = (SV )s2
mean that
πij r (V ) = 0
if j gt s1 or r gt s2 In particular V belongs to
dimVoplusi=1
A(i )s1+s2
Hence V =Vs1+s2 Similarly using
(B otimesC )(n) = B(n) otimesC(n)
one shows V =V(maxt1t2)
Proposition 47 [39 Proof of Lemma 45] Let G be a reductive LDAG S be the differential
commutator subgroup of G (ie the Kolchin-closure of the commutator subgroup of G) and T
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 33
be the identity component of the center of G The LDAG S is semisimple and the multiplication
map
micro S timesT rarrG (s t ) 7rarr st
is an epimorphism of LDAGs with a finite kernel
Let Rep(n) G denote the tensor subcategory of RepG generated by P n(W ) (the nth total
prolongation) The following Proposition shows that Rep(n) G does not depend on the choice of
W
Proposition 48 For all V isin RepG V isin Rep(n) G if and only if V =V(n)
Proof Suppose V isin Rep(n) G Since the matrix entries of P n(W ) belong to A(n) we have V =V(n)
Conversely suppose V = V(n) Then V is a representation of the LAG G(n) whose Hopf algebra
is A(n) Since P n(W ) is a faithful A-comodule it is a faithful A(n)-comodule Hence RepG(n) is
generated by P n(W )
If τ(G) Eacute 0 then by [41 Section 321] there exists n isinN such that
RepG = langRep(n) G
rangotimes
The smallest such n will be denoted by ord(G) For a G-module V let ``(V ) denote the length of
the socle filtration of V In particular we have
``(V ) Eacute dimV
For a G-module V let ``(V ) denote the length of the socle filtration of V In particular we
have
``(V ) Eacute dimV
Theorem 49 Let G be a reductive LDAG with τ(Z (G)
)Eacute 0 and T = Z (G) For all V isin RepG we
have V isin Rep(n) G where
n = max``(V )minus1ord(T ) (47)
Proof Let V isin RepG By Proposition 48 we need to show that V =V(n) where n is given by (47)
Set G = StimesT where S subG is the differential commutator subgroup of G The multiplication map
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
34 A Minchenko et al
micro G rarrG (see Proposition 47) induces the structure of a G-module on the space V which we will
denote by V By Theorem 45
SV = SVr = SV(r )
where
r = ``(SV
)minus1 = ``(SV )minus1
It follows from Proposition 312 (formula (318)) and Lemma 35 that if W isin RepG is semisimple
then SW isin RepS is semisimple Hence
``(SV ) Eacute ``(V )
Therefore
SV =S V(s) s = ``(V )minus1
Next since τ(T ) Eacute 0 we have
RepT = Rep(t ) T t = ord(T )
By Proposition 48 T V = T V(t ) Proposition 46 implies
V = V(maxst ) = V(n)
Now applying Proposition 312 to φ =micro we obtain V =V(n)
The following proposition suggests an algorithm to find ord(T )
Proposition 410 Let G sub GL(W ) be a reductive LDAG with τ(Z (G)
) Eacute 0 where the G-module
W is semisimple Set T = Z (G) and H =G sub GL(W ) Let
H rarr GL(U )
be an algebraic representation with Ker = [H H] Then ord(T ) is the minimal number t such
that the differential tensor category generated by GU isin RepG coincides with the tensor category
generated by P t (GU ) isin RepG
Proof We have (G) = (T ) and Ker cap T is finite Propositions 312 and 48 complete the
proof
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 35
5 Computing parameterized differential Galois groups
In this section we show how the main results of the paper can be applied to constructing
algorithms that compute the maximal reductive quotient of a parameterized differential Galois
group and decide if a parameterized Galois is reductive
51 Linear differential equations with parameters and their Galois theory
In this section we will briefly recall the parameterized differential Galois theory of linear
differential equations also known as the PPV theory [14] Let K be a ∆prime = partpart1 partm-field and
partY = AY A isin Mn(K ) (51)
be a linear differential equation (with respect to part) over K A parameterized PicardndashVessiot
extension (PPV-extension) F of K associated with (51) is a ∆prime-field F sup K such that there exists
a Z isin GLn(F ) satisfying partZ = AZ F part = K part and F is generated over K as a ∆prime-field by the entries of
Z (ie F = K langZ rang)
The field K part is a ∆ = part1 partm-field and if it is differentially closed a PPV-extension
associated with (51) always exists and is unique up to a ∆prime-K -isomorphism [14 Proposition 96]
Moreover if K part is relatively differentially closed in K then F exists as well [21 Thm 25] (although
it may not be unique) Some other situations concerning the existence of K have also been treated
in [60]
If F = K langZ rang is a PPV-extension of K one defines the parameterized PicardndashVessiot Galois
group (PPV-Galois group) of F over K to be
G = σ F rarr F |σ is a field automorphism σδ= δσ for all δ isin∆prime and σ(a) = a a isin K
For anyσ isinG one can show that there exists a matrix [σ]Z isin GLn(K part
)such thatσ(Z ) = Z [σ]Z and
the map σ 7rarr [σ]Z is an isomorphism of G onto a differential algebraic subgroup (with respect to
∆) of GLn(K part
)
One can also develop the PPV-theory in the language of modules A finite-dimensional
vector space M over the ∆prime-field K together with a map part M rarr M is called a parameterized
differential module if
part(m1 +m2) = part(m1)+part(m2) and part(am1) = part(a)m1 +apart(m1) m1m2 isin M a isin K
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
36 A Minchenko et al
Let e1 en be a K -basis of M and ai j isin K be such that part(ei ) = minussumj a j i e j 1 Eacute i Eacute n As in [57
Section 12] for v = v1e1 + + vnen
part(v) = 0 lArrrArr part
v1
vn
= A
v1
vn
A = (ai j )ni j=1
Therefore once we have selected a basis we can associate a linear differential equation of the
form (51) with M Conversely given such an equation we define a map
part K n rarr K n part(ei ) =minussumj
a j i e j A = (ai j )ni j=1
This makes K n a parameterized differential module The collection of parameterized differential
modules over K forms an abelian tensor category In this category one can define the notion of
prolongation M 7rarr Pi (M) similar to the notion of prolongation of a group action as in (21) For
example if partY = AY is the differential equation associated with the module M then (with respect
to a suitable basis) the equation associated with Pi (M) is
partY =(
A parti A
0 A
)Y
Furthermore if Z is a solution matrix of partY = AY then
(Z parti Z
0 Z
)
satisfies this latter equation Similar to the s th total prolongation of a representation we define
the s th total prolongation P s (M) of a module M as
P s (M) = P s1P s
2 middot middotP sm(M)
If F is a PPV-extension for (51) one can define a K part-vector space
ω(M) = Ker(part M otimesK F rarr M otimesK F )
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 37
The correspondence M 7rarr ω(M) induces a functor ω (called a differential fiber functor) from
the category of differential modules to the category of finite-dimensional vector spaces over K part
carrying Pi rsquos into the Pi rsquos (see [21 Defs 49 422] [45 Definition 2] [29 Definition 427] [28
Definition 412] for more formal definitions) Moreover
(RepG forget
) sim=(langP i1
1 middot middotP imm (M) | i1 im Ecirc 0rangotimesω
)(52)
as differential tensor categories [21 Thms 427 51] This equivalence will be further used in the
rest of the paper to help explain the algorithms
In Section 53 we shall restrict ourselves to PPV-extensions of certain special fields We now
describe these fields and give some further properties of the PPV-theory over these fields Let K(x)
be the ∆prime = partpart1 partm-differential field defined as follows
(i) K is a differentially closed field with derivations ∆= part1 partm
(ii) x is transcendental over K and (53)
(iii) parti (x) = 0 i = 1 m part(x) = 1 and part(a) = 0 for all a isin K
When one further restricts K Proposition 51 characterizes the LDAGs that appear as PPV-Galois
groups over such fields We say that K is a universal differential field if for any differential field k0 subK differentially finitely generated over Q and any differential field k1 sup k0 differentially finitely
generated over k0 there is a differential k0-isomorphism of k1 into K ([33 Chapter IIISection 7])
Note that a universal differential field is differentially closed
Proposition 51 (cf [19 42]) Let K be a universal ∆-field and let K(x) satisfy conditions (53) An
LDAG G is a parameterized differential Galois group over K(x) if and only if G is a DFGG
Assuming that K is only differentially closed one still has the following corollary
Corollary 52 Let K(x) satisfy conditions (53) If G is reductive and is a parameterized differential
Galois group over K(x) then τ(Z (G)) Eacute 0
Proof Let L be a PPV-extension of K(x) with parameterized differential Galois group G and let
U be a universal differential field containing K (such a field exists [33 Chapter IIISection 7])
Since K is a fortiori algebraically closed U otimesKL is a domain whose quotient field we denote by
U L One sees that the ∆-constants C of U L are U We may identify the quotient field U (x) of
U otimesKK(x) with a subfield of U L and one sees that U L is a PPV-extension of U (x) Furthermore
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
38 A Minchenko et al
the parameterized differential Galois group of U L over U (x) is G(U ) (see also [21 Section 8])
Proposition 51 implies that G(U ) is a DFGG Lemma 214 implies that
tr degU Ulang
Z(G)rangltinfin
Since G is defined over K and K is algebraically closed tr degKKlang
Z (G)rang lt infin Therefore
τ(Z (G)) Eacute 0
52 Equivalent statements of reductivity
In this section we give a characterization of parameterized differential modules whose PPV-
Galois groups are reductive LDAGs which will be used in Section 53 to construct the main
algorithms
In this section let K be a differential field as at the beginning of Section 51 Given a
parameterized differential module M such that it has a PPV-extension over K let G be its
PPV-Galois group Recall a construction of the ldquodiagonal partrdquo of M denoted by Mdiag which
induces [45] a differential representation
ρdiag G rarr GL(ω
(Mdiag
))
where ω is the functor of solutions If M is irreducible we set Mdiag = M Otherwise if N is a
maximal differential submodule of M we set
Mdiag = Ndiag oplusMN
Since M is finite-dimensional and dim N lt dim M Mdiag is well-defined above Another descrip-
tion of Mdiag is let
M = M0 sup M1 sup sup Mr = 0 (54)
be a complete flag of differential submodules that is Miminus1Mi are irreducible We then let
Mdiag =roplus
i=1Miminus1Mi
A version of the JordanndashHoumllder Theorem implies that Mdiag is unique up to isomorphism Note
that Mdiag is a completely reducible differential module The complete flag (54) corresponds to a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 39
differential equation in block upper triangular form
partY =
Ar
0 Arminus1
0 0 A2
0 0 0 A1
Y (55)
where for each matrix Ai the differential module corresponding to partY = Ai Y is irreducible The
differential module Mdiag corresponds to the block diagonal equation
partY =
Ar 0 0
0 Arminus1 0 0
0 0 A2 0
0 0 0 A1
Y (56)
Furthermore given a complete flag (54) we can identify the solution space of M in the
following way Let V be the solution space of M and
V =V0 supV1 sup supVr = 0 (57)
be a complete flag of spaces of V where each Vi is the solution space of Mi Note that each Vi is a
G-submodule of V and that all Vi Vi+1 are simple G-modules One then sees that
Vdiag =roplus
i=1Viminus1Vi
Proposition 53 Let
micro G rarrG
Ru
(G
)rarrG sub GL(ω(M))
be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
40 A Minchenko et al
Proof Since ρdiag is completely reducible ω(Mdiag
)is a completely reducible ρdiag
(G
)-module
Therefore ρdiag
(G
)is a reductive LAG [54 Chapter 2] Hence
Ru
(G
)sub Kerρdiag
where ρdiag is considered as a map from G On the other hand by definition Kerρdiag consists of
unipotent elements only Therefore since Kerρdiag is a normal subgroup of GM and connected by
[59 Corollary 85]
Kerρdiag = Ru
(G
) (58)
Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru
(GM
)) [25 Theo-
rem VIII43] (58) implies that ρdiag is equivalent to micro
Corollary 54 In the notation of Proposition 53 ρdiag is faithful if and only if
G rarrGRu
(G
)(59)
is injective
Proof Since ρdiagsim= micro by Proposition 53 faithfulness of ρdiag is equivalent to that of micro which is
precisely the injectivity of (59)
Proposition 55 The following statements are equivalent
1 ρdiag is faithful
2 G is a reductive LDAG
3 there exists q Ecirc 0 such that
M isin langP q (
Mdiag)rang
otimes (510)
Proof (1) implies (3) by [44 Proposition 2] and [45 Corollary 3 and 4] If a differential representa-
tion micro of the LDAG G is not faithful so are the objects in the categorylang
P q (micro)rangotimes for all q Ecirc 0 Using
the equivalence of neutral differential Tannakian categories from [45 Theorem 2] this shows
that (3) implies (1)
If ρdiag is faithful then G is reductive by the first part of the proof of [39 Theorem 47]
showing that (1) implies (2) Suppose now that G is a reductive LDAG Since Ru
(G
)capG is a
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 41
connected normal unipotent differential algebraic subgroup of G it is equal to id Thus (59)
is injective and by Corollary 54 (2) implies (1)
53 Algorithm
In this section we will assume that K(x) satisfies conditions (53) and furthermore that K
is computable that is one can effectively carry out the field operations and effectively apply
the derivations We will describe an algorithm for calculating the maximal reductive quotient
GRu(G) of the PPV- Galois group G of any partY = AY A isin GLn(K(x)) and an algorithm to decide if
G is reductive that is if G equals this maximal reductive quotient
531 Ancillary Algorithms
We begin by describing algorithms to solve the following problems which arise in our two main
algorithms
(A) Let K be a computable algebraically closed field and H sub GLn(K ) be a reductive LAG
defined over K Given the defining equations for H find defining equations for H and Z (H)
as well as defining equations for normal simple algebraic groups H1 H` of H such that the
homomorphism
π H1 times timesH`timesZ (H) rarr H
is surjective with a finite kernel [20] gives algorithms for finding Groumlbner bases of the radical of
a polynomial ideal and of the prime ideals appearing in a minimal decomposition of this ideal
Therefore one can find the defining equations of H Elimination properties of Groumlbner bases
allow one to compute
Z (H) = h isin H | g hgminus1 = h for all g isin H
We may write H = S middot Z (H) where S = [H H] is semisimple A theorem of Ree [46] states that
every element of a connected semisimple algebraic group is a commutator so
S = [h1h2] |h1h2 isin H
Using the elimination property of Groumlbner bases we see that one can compute defining equations
for S We know that S = H1 middot middotH` for some simple algebraic groups Hi We now will find the Hi
Given the defining ideal J of S the Lie algebra s of S is
s isin Mn(K ) | f (In +εs) = 0 mod ε2 for all f isin J
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
42 A Minchenko et al
where ε is a new variable This K -linear space is also computable via Groumlbner bases techniques
In [16 Section 115] one finds algorithms to decide if s is simple and if not how to decompose s
into a direct sum of simple ideals s= s1oplus opluss` Note that each si is the tangent space of a normal
simple algebraic subgroup Hi of S and S = H1 middot middotH` Furthermore H1 is the identity component
of
h isin S | Ad(h)(s2 oplus opluss`) = 0
and this can be computed via Groumlbner bases methods Let S1 be the identity component of
h isin S | Ad(h)(s1) = 0
We have S = H1 middot S1 and we can proceed by induction to determine H2 H` such that S1 =H2 middot middotH` The groups Z (H) and H1 H` are what we desire
(B) Given A isin Mn(K(x)) find defining equations for the PV-Galois group H sub GLn(K) of the
differential equation partY = AY When H is finite construct the PV-extension associated with this
equation A general algorithm to compute PV-Galois groups is given by Hrushovski [26] When
H is assumed to be reductive an algorithm is given in [15] An algorithm to find all algebraic
solutions of a differential equation is classical (due to Painleveacute and Boulanger) and is described
in [47 48]
(C) Given A isin Mn(K(x)) and the fact that the PPV-Galois group G of the differential equation
partY = AY satisfies τ(G) Eacute 0 find the defining equations of G An algorithm to compute this is given
in [41 Algorithm 1]
(D) Assume that we are given an algebraic extension F of K(x) a matrix A isin Mn(F ) the defining
equations for the PV-Galois group G of the equation partY = AY over F and the defining equations for
a normal algebraic subgroup H of G Find an integer ` a faithful representation ρ GH rarr GL`(K)
and a matrix B isin M`(F ) such that the equation partY = BY has PV-Galois group ρ(GH)
The usual proof ([27 Section 115]) that there exists an ` and a faithful rational representa-
tion ρ GH rarr GL`(K) is constructive that is if V Kn is a faithful G-module and we are given the
defining equations for G and H then using direct sums subquotients duals and tensor prod-
ucts one can construct a G-module W K` such that the map ρ G rarr GL`(K) has kernel H
Let M be the differential module associated with partY = AY Applying the same constructions
to M yields a differential module N The Tannakian correspondence implies that the action of G
on the associated solution space is (conjugate to) ρ(G)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 43
(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n
Let
W = (Z c1 c`) | Z isin F n c1 c` isin K and partZ + AZ = c1B1 + + c`B`
Find a K-basis of W Let F [part] be the ring of differential operators with coefficients in F Let
C = Inpart+ A isin Mn(F [part])
We may write partZ + AZ = c1B1 + + c`B` as
C Z = c1B1 + + c`B`
Since F [part] has a left and right division algorithm ([57 Section 21]) one can row and column
reduce the matrix C that is find a left invertible matrix U and a right invertible matrix V
such that UCV = D is a diagonal matrix We then have that (Z c1 c`) isin W if and only if
X = (V minus1Z c1 c`) satisfies
D X = c1U B1 + + c`U B`
Since D is diagonal this is equivalent to finding bases of scalar parameterized equations
Ly = c1b1 + + c`b` L isin F [part] bi isin K
[49 Proposition 31 and Lemma 32] give a method to solve this latter problem We note that if
A isin K(x) and `= 1 an algorithm for finding solutions with entries in K(x) directly without having
to diagonalize is given in [3]
(F) Let A isin Mn(K(x)) and let M be the differential module associated with partY = AY Find a basis
of M so that the associated differential equation partY = BY B isin Mn(K(x)) is as in (55) that is in
block upper triangular form with the blocks on the diagonal corresponding to irreducible modules
We are asking to ldquofactorrdquo the system partY = AY Using cyclic vectors one can reduce this problem
to factoring linear operators of order n for which there are many algorithms (cf [57 Section 42])
A direct method is also given in [23]
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
44 A Minchenko et al
(G) Suppose that we are given F an algebraic extension of K(x) A isin Mn(F ) and the defining
equations of the PV-Galois group H of partY = AY Assuming that H is a simple LAG find the PPV-
Galois G group of partY = AY Let D be the K-span of ∆ A Lie K-subspace E of D is a K-subspace
such that if DD prime isin E then
[DD prime] = DD primeminusD primeD isin E
We know that the group G is a Zariski-dense subgroup of H The Corollary to [13 Theorem 17]
states that there is a Lie K-subspace E sub D such that G is conjugate to H(KE
) Therefore to
describe G it suffices to find E Let
W = (Z c1 cm) | Z isin Mn(F ) = F n2
c1 cm isin K and partZ + [Z A] = c1part1 A+ + cmpartm A
The algorithm described in (E) allows us to calculate W We claim that we can take
E = c1part1 + + cmpartm | there exists Z isin GLn(F ) such that (Z c1 cm) isinW
(511)
Note that this E is a Lie K-subspace of D To see this it suffices to show that if D1D2 isin E then
[D1D2] isin E If
partB1 + [B1 A] = D1 A and partB2 + [B2 A] = D2 A for some B1B2 isin GLn(F )
then a calculation shows that
partB + [B A] = [D1D2]A where B = D1B2 minusD2B1 minus [B1B2]
In particular [34 Section 05 Propostions 6 and 7] imply that E has a K-basis of commuting
derivationspart1 partt
that extends to a basis of commuting derivations
part1 partm
of D
To show that G is conjugate to H(KE
)we shall need the following concepts and results
Let ∆prime =
partpart1 partm
and k be a ∆prime-field Let ∆ =
part1 partm
and Σ sub ∆ Assume that C = kpart
is differentially closed
Definition 56 Let A isin M(k) We say partY = AY is integrable with respect to Σ if for all parti isinΣ there
exists Ai isin Mn(k) such that
partA j minuspart j A = [A A j ] for all part j isinΣ and (512)
parti A j minuspart j Ai = [Ai A j ] for all parti part j isinΣ (513)
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 45
The following characterizes integrability in terms of the behavior of the PPV-Galois group
Proposition 57 Let K be the PPV-extension of k for partY = AY and let G sub GLn(C ) be the PPV-
Galois group The group G is conjugate to a subgroup of GLn(CΣ
)if and only if partY = AY is
integrable with respect to Σ
Proof Assume that G is conjugate to a subgroup of GLn(CΣ
)and let B isin GLn(C ) satisfy
BGBminus1 sub GLn(CΣ
)
Let Z isin GLn(K ) satisfy partZ = AZ and W = Z Bminus1 For any V isin GLn(K ) such that partV = AV andσ isinG
we will denote by [σ]V the matrix in GLn(C ) such that σ(V ) =V [σ]V We have
σ(W ) = Z [σ]Z Bminus1 = Z Bminus1B [σ]Z Bminus1 =W [σ]W
so
[σ]W = B [σ]Z Bminus1 isin GLn(CΣ
)
A calculation shows that Ai = parti W middotW minus1 is left fixed by all σ isinG and so lies in Mn(k) Since the parti
commute with part and each other we have that the Ai satisfy (512) and (513)
Now assume that partY = AY is integrable with respect to Σ and for convenience of notation
let Σ= part1 partt
We first note that since C is differentially closed with respect to ∆ the field CΣ
is differentially closed with respect to Π = partt+1 partm
(in fact CΣ is also differentially closed
with respect to ∆ see [37]) Note that CΣ = kpartcupΣ Let
R = kZ 1(det Z )∆prime
be the PPV-extension ring of k for the integrable system
partY = AY (514)
parti Y = Ai Y i = 1 t (515)
The ring R is a ∆prime-simple ring generated both as a Π-differential ring and as a ∆-differential ring
by the entries of Z and 1det Z Therefore R is also the PPV-ring for the single equation (514)
([24 Definition 610])
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
46 A Minchenko et al
Let L be the quotient field of R The group G of ∆prime-automorphisms of L over k is both the
PPV-group of the system (514) (515) and of the single equation (514) In the first case we see
that the matrix representation of this group with respect to Z lies in GLn(CΣ
)and therefore the
same is true in the second case Since CΣ is differentially closed the PPV-extension K = klangU rang is k-
isomorphic to L as ∆prime-fields This isomorphism will take U to Z D for some D isin GLn(C ) and so the
matrix representation of the PPV-group of K over k will be conjugate to a subgroup of GLn(CΣ
)
One can also argue as follows First note that C is alsoΣ-differentialy closed by [37] For every
∆-LDAG G prime sub GLn(C ) with defining ideal
I subC Xi j 1det∆
let G primeΣ denote the Σ-LDAG with defining ideal
J = I capC Xi j 1detΣ
Then G prime is conjugate to Σ-constants if and only if G primeΣ is Indeed the former is equivalent to the
existence of D isin GLn(C ) such that for all i j 1 Eacute i j Eacute n and part isin Σ we have part(D Xi j Dminus1
)i j isin I
which holds if and only if part(D X Dminus1
)i j isin J
Let K = klangZ rang∆prime The Σ-field KΣ = klangZ rangpartcupΣ is a Σ-PPV extension for partY = AY by definition
As in [14 Proposition 36] one sees that GΣ is its Σ-PPV Galois group Finally GΣ is conjugate to
Σ-constants if and only if partY = AY is integrable with respect to Σ by [14 Proposition 39]
Corollary 58 Let K be the PPV-extension of k for partY = AY and G sub GLn(C ) be the PPV-Galois
group Then G is conjugate to a subgroup of GLn(CΣ
)if and only if for every parti isin Σ there exists
Ai isin Mn(k) such that partA j + [A j A] = part j A
Proof In [22 Theorem 44] the authors show that G is conjugate to a subgroup of GLn(CΣ
)if and only if for each parti isin Σ G is conjugate to a subgroup of GLn
(Cparti
) Two applications of
Proposition 57 yields the conclusion
Applying Corollary 58 to part= part and the commuting basis Σ= part1 partt
of E implies that G
is conjugate to H(KE
)
Sections 532 and 533 now present the two algorithms described in the introduction
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 47
532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group
G
Assume that we are given a matrix A isin Mn(K) Let H be the PV-Galois group of this equation We
proceed as follows taking into account the following general principle For every normal algebraic
subgroup H prime of H and B isin M`(K) if HH prime is the PV-Galois group of partY = BY then G(G capH prime) is
its PPV-Galois group which follows from (D)
Step 1 Reduce to the case where H is reductive Using (F) we find an equivalent differential equa-
tion as in (55) whose matrix is in block upper triangular form where the modules corresponding
to the diagonal blocks are irreducible We now consider the block diagonal Equation (56) This
latter equation has PPV-Galois group GRu(G)
Step 2 Reduce to the case where G is connected and semisimple We will show that it is sufficient
to be able to compute the PPV-Galois group of an equation partY = AY assuming A has entries in an
algebraic extension of K(x) assuming we have the defining equations of the PV-Galois group of
partY = AY and assuming this PV-Galois group is connected and semisimple
Using (B) we compute the defining equations of the PV-Galois group H of partY = AY over
K(x) Using (A) we calculate the defining equations for H and Z(H) as well as defining
equations for normal simple algebraic groups H1 H` of H as in (A) Note that
H = SH middotZ(H)
where SH = H1 middot middotH` is the commutator subgroup of H Note that
SG = [GG]
is Zariski-dense in SH Using (D) we construct a differential equation partY = BY whose PV-Galois
group is HH This latter group is finite so this equation has only algebraic solutions and again
using (B) we can construct a finite extension F of K(x) that is the PV-extension corresponding to
partY = BY The PV-Galois group of partY = AY over F is H
Since we have the defining equations of Z (H) (D) allows us to construct a representation
ρ H rarr HZ(H)
and a differential equation partY = BY B having entries in F whose PV-Galois group is ρ(H) Note
that ρ(G) is the PPV-Galois group of partY = BY and is Kolchin-dense in ρ
(H) Therefore ρ
(G) is
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
48 A Minchenko et al
connected and semisimple Let us assume that we can find defining equations of ρ(G) We can
therefore compute defining equations of ρminus1(ρ(G)) The group
ρminus1(ρ(G))capSH
normalizes[GG] in SH By Lemma 59 we have
ρminus1(ρ(G))capSH = SG
Therefore we can compute the defining equations of SG
To compute the defining equations of G we proceed as follows Using (D) we compute a
differential equation partY = BY B having entries in K(x) whose PV-group is HSH The PPV-Galois
group of this equation is L =GSG By Lemma 214 this group has differential type at most 0 so (C)
implies that we can find the defining equations of L Let
ρ H rarr HSH
We claim that
G = ρminus1(L)capNH(SG
)
Clearly
G sub ρminus1(L)capNH(SG
)
Now let
h isin ρminus1(L)capNH(SG
)
We can write h = h0g where g isin G and h0 isin SH Furthermore h0 normalizes SG Lemma 59
implies that h0 isin SG and so h isin G Since we can compute the defining equations of SG we can
compute the defining equations of NH (SG ) Since we can compute ρ and the defining equations
of L we can compute the defining equations of ρminus1(L) and so we get the defining equations of G
All that remains is to prove the following lemma
Lemma 59 Let G be a Zariski-dense differential subgroup of a semisimple linear algebraic group
H Then
1 Z (H) subG and
2 NH (G) =G
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 49
Proof [13 Theorem 15] implies that
H = H1 middot middotH` and G =G1 middot middotG`
where each Hi is a normal simple algebraic subgroup of H with [Hi H j ] = 1 for i 6= j and each Gi
is Zariski-dense in Hi and normal in G Therefore it is enough to prove the claims when H itself is
a simple algebraic group In this case let us assume that H sub GL(V ) where H acts irreducibly on
V Schurrsquos Lemma implies that the center of H consists of scalar matrices and since H = (H H)
these matrices have determinant 1 Therefore the matrices are of the form ζI where ζ is a root of
unity [13 Theorem 19] states that there is a Lie K -subspace E of D the K-span of ∆ such that G
is conjugate to H(KE
) Since the roots of unity are constant for any derivation we have that the
center of H lies in G
To prove NH (G) =G assume G = H(KE
)and let g isinG and h isin NH (G) For any part isin E we have
0 = part(hminus1g h)=minushminus1part(h)hminus1g h +hminus1gpart(h)
Therefore part(h)hminus1 commutes with the elements of G and so must commute with the elements
of H Again by Schurrsquos Lemma part(h)hminus1 is a scalar matrix On the other hand part(h)hminus1 lies in the
Lie algebra of H ([33 Section V22 Proposition 28]) and so the trace of part(h)hminus1 is zero Therefore
part(h)hminus1 = 0 Since part(h) = 0 for all part isin E we have h isinG
Step 3 Computing G when G is connected and semisimple We have reduced the problem to
calculating the PPV-Galois group G of an equation partY = AY where the entries of A lie in an
algebraic extension F of K(x) and where we know the equations of the PV-Galois H group of this
equation over F Let
H = H1 middot middotH` and G =G1 middot middotG`
where the Hi are simple normal subgroups of H and Gi is Zariski-dense in Hi Using (D) we
construct for each i an equation partY = Bi Y with Bi isin Mn(F ) whose PV-Galois group is HHi
where
Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint
[51] M Singer and F Ulmer Galois groups of second and third order linear differential equations Journal
of Symbolic Computation 16(3)9ndash36 1993 URL httpdxdoiorg101006jsco19931032
[52] M Singer and F Ulmer Liouvillian and algebraic solutions of second and third order linear differential
equations Journal of Symbolic Computation 16(3)37ndash73 1993 URL httpdxdoiorg101006
jsco19931033
[53] M Singer and F Ulmer Necessary conditions for Liouvillian solutions of (third order) linear differential
equations Applied Algebra in Engineering Communication and Computing 6(1)1ndash22 1995 URL
httpdxdoiorg101007BF01270928
[54] T A Springer Invariant theory Springer-Verlag Berlin-New York 1977 URL httpdxdoiorg
101007BFb0095644
[55] M Sweedler Hopf algebras W A Benjamin New York 1969
[56] F Ulmer and J-A Weil A note on Kovacicrsquos algorithm Journal of Symbolic Computation 22(2)179ndash200
1996 URL httpdxdoiorg101006jsco19960047
[57] M van der Put and M Singer Galois theory of linear differential equations Springer Berlin 2003 URL
httpdxdoiorg101007978-3-642-55750-7
[58] M van Hoeij J-F Ragot F Ulmer and J-A Weil Liouvillian solutions of linear differential equations
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 61
of order three and higher Journal of Symbolic Computation 28(4-5)589ndash610 1999 URL http
dxdoiorg101006jsco19990316
[59] W Waterhouse Introduction to Affine Group Schemes Springer Berlin 1979 URL httpdxdoi
org101007978-1-4612-6217-6
[60] M Wibmer Existence of part-parameterized PicardndashVessiot extensions over fields with algebraically
closed constants Journal of Algebra 361163ndash171 2012 URL httpdxdoiorg101016j
jalgebra201203035
Introduction
Basic definitions
Differential algebra
Linear Differential Algebraic Groups
Representations of LDAGs
Unipotent radical of differential algebraic groups and reductive LDAGs
Differentially finitely generated groups
Filtrations and gradings of the coordinate ring of an LDAG
Filtrations of G-modules
Subalgebras generated by W-filtrations
Filtrations of G-modules in reductive case
Socle of a G-module
Main result for semisimple LDAGs
Reductive case
Computing parameterized differential Galois groups
Linear differential equations with parameters and their Galois theory
Equivalent statements of reductivity
Algorithm
Ancillary Algorithms
An algorithm to compute the maximal reductive quotient G`39`42`613A``45`47`603ARu(G) of a PPV-Galois group G
An algorithm to decide if the PPV-Galois group of a parameterized linear differential equation is reductive
Examples
50 A Minchenko et al
and a surjective homomorphism πi H rarr HHi Note that HHi is a connected simple LAG
Therefore (G) allows us to calculate the PPV-Galois group Gi of partY = Bi Y We claim that
Gi =πminus1i
(Gi
)capHi
To see this note that Hi capHi lies in the center of Hi and therefore must lie in Gi by Lemma 59
Therefore we have defining equations for each Gi and so can construct defining equations for
G
533 An algorithm to decide if the PPV-Galois group of a parameterized linear differential
equation is reductive
Let K(x) be as in (53) Assume that we are given a differential equation partY = AY with A isinMn(K(x)) Using the solution to (F) above we may assume that A is in block upper triangular form
as in (55) with the blocks on the diagonal corresponding to irreducible differential modules Let
Adiag be the corresponding diagonal matrix as in (56) let M G and MdiagGdiag be the differential
modules and PPV-Galois groups associated with partY = AY and partY = AdiagY respectively Of
course
Gdiag GRu(G)
so G is reductive if and only if Gdiag G
This implies via the Tannakian equivalence that the differential tensor category generated by
Mdiag is a subcategory of the differential tensor category generated by M and that G is reductive if
and only if these categories are the same The differential tensor category generated by a module
M is the usual tensor category generated by all the total prolongations P s (M) of that module
From this we see that G is a reductive LDAG if and only if M belongs to the tensor category
generated by some total prolongation P s (Mdiag) Therefore to decide if G is reductive it suffices
to find algorithms to solve problems (H) and (I) below
(H) Given differential modules M and N decide if M belongs to the tensor category generated
by N Since we are considering the tensor category and not the differential tensor category
this is a question concerning nonparameterized differential equations Let KN KM KMoplusN be PV-
extensions associated with the corresponding differential modules and let GM GN GNoplusM be the
corresponding PV-Galois groups The following four conditions are easily seen to be equivalent
(a) N belongs to the tensor category generated by M
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 51
(b) KN sub KM considered as subfields of KMoplusN
(c) KMoplusN = KM
(d) the canonical projection π GMoplusN subGM oplusGN rarrGM is injective (it is always surjective)
Therefore to solve (H) we apply the algorithmic solution of (B) to calculate GMoplusN and GM and
using Groumlbner bases decide if π is injective
(I) Given M and Mdiag as above calculate an integer s such that if M belongs to the differential
tensor category generated by Mdiag then M belongs to the tensor category generated by P s(Mdiag
)
We will apply Theorem 49 and Proposition 410 Note that since the PPV-Galois group Gdiag
associated to Mdiag is reductive Lemma 214 implies that we may apply these results to Gdiag
Theorem 49 implies that such a bound is given by the integer
max``(V )minus1ord(T )
where V is a solution space associated with Mdiag and T = Z(G
diag
) As noted in the discussion
preceding Theorem 49
``(V ) Eacute dimK(V ) = dimK(x) Mdiag
Proposition 410 implies that ord(T ) can be bounded in the following way Using the algorithm to
solve (B) we calculate the defining equations of the PV-Galois group Hdiag associated with Mdiag
and then calculate the defining equations of Hdiag and
[H
diag Hdiag
](as in (A)) Using the solution
to (D) one calculates a differential equation partY = BY whose PV-Galois group is
H[
Hdiag H
diag
]
Denote the associated differential module by N Proposition 410 implies that ord(T ) is the
smallest value of t so that the differential tensor category generated by N coincides with the tensor
category generated by P t (N ) The following conditions are easily seen to be equivalent
(a) The differential tensor category generated by N coincides with the tensor category gener-
ated by P t (N )
(b) The tensor category generated by P t (N ) coincides with the tensor category generated by
P t+1(N )
(c) P t+1(N ) belongs to the tensor category generated by P t (N )
Therefore to bound ord(T ) one uses the algorithm of (H) to check for t = 012 if P t+1(N )
belongs to the tensor category generated by P t (N ) until this event happens (see also [41
52 A Minchenko et al
Section 321 Algorithm 1]) As noted in the discussion preceding Theorem 49 this procedure
eventually halts Taking the maximum of this t and dimK(x) M minus1 yields the desired s
6 Examples
In this section we will illustrate both Theorem 45 and our main algorithm In Example 62 we
will show that the bound in Theorem 45 is sharp Example 63 is an illustration of the algorithm
Example 61 Following [40 Ex 418] let
V = spanK1 x prime
11x21 minusx11x prime21 x prime
11x22 minusx prime21x12 x prime
12x22 minusx12x prime22 x prime
11x22 minusx prime12x21
sub A
where
A = Kx11 x12 x21 x22
[x11x22 minusx12x21 minus1] (61)
which induces the following differential representation of SL2
SL2(U ) 3(
a b
c d
)7rarr
1 aprimec minusac prime aprimed minusbc prime bprimed minusbd prime aprimed primeminusbprimec prime
0 a2 ab b2 abprimeminusaprimeb
0 2ac ad +bc 2bd 2(ad primeminusbc prime)
0 c2 cd d 2 cd primeminus c primed
0 0 0 0 1
under the right action of SL2 on A Since the length of the socle filtration for V is 3 let n = 2
Theorem 45 claims that V isin langP 2
(Vdiag
)rangotimes We will show that in fact
V isin langP
(Vdiag
)rangotimes (62)
Indeed by the ClebschndashGordon formula for tensor products of irreducible representations of SL2
the usual irreducible representation U = spanKu v of SL2 is a direct summand of Vdiag otimesVdiag
Moreover
V sub (P (U )oplusP (U ))otimes (P (U )oplusP (U ))
under the embedding
U oplusU rarr A (au +bvcu +d v) 7rarr ax11 +bx12 + cx21 +d x22
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 53
which implies (62)
Example 62 Consider the first prolongations P (V ) of the usual (irreducible) representation
r SL2 rarr GL(V ) of dimension 2
P (r ) SL2 3 A 7rarr(
A Aprime
0 A
)
The length of the socle filtration is 2 and we tautologically have
P (V ) isin langP 2minus1 (
P (V )diag)rang
otimes
Note that
P (V ) notin langP (V )diag
rangotimes
as every object oflang
P (V )diagrangotimes = langV rangotimes is completely reducible [39 Thm 47] but P (V ) is not
completely reducible [44 Proposition 3] [22 Theorem 46] By Proposition 44 for all n Ecirc 0
P n(V )n sub socn+1 P n(V ) (63)
Since ror = ρ V rarrV otimesK A0 where A is defined in (61) for all n Ecirc 0
P n(ρ) P n(V ) rarr P n(V )otimesK An
(see (33)) Therefore P n(V )n = P n(V ) Since P n(V ) sup socn+1 P n(V ) (63) implies that
P n(V ) = socn+1 P n(V )
Therefore the length of the socle filtration of P n(V ) does not exceed n +1 If
P n+1(V ) isin langP n(V )
rangotimes (64)
then for all q gt n P q (V ) isin langP n(V )rangotimes which implies that
langP i (V ) | i Ecirc 0
rangotimes = lang
P n(V )rangotimes (65)
54 A Minchenko et al
By [17 Proposition 220] (65) implies that A is a finitely generated K-algebra which is not the
case Therefore (64) does not hold Thus the bound in Theorem 45 is sharp
We will now illustrate how the algorithm works Let C denote the differential closure of
Q with respect to a single derivation partt In the following examples we consider the differential
equations over the field K(x) =C (x) with derivations ∆prime = partx partt and ∆= partt
Example 63 As in [41 Ex 34] consider the equation partx Y = AY where
A =(
1 tx + 1
x+1
0 1
)
whose PV-group is (a b
0 a
) ∣∣∣ a b isinU a 6= 0
Gm timesGa (66)
which is not reductive Let M be the corresponding differential module Using our algorithm we
will test whether the PPV-Galois group G of partx Y = AY is reductive We have
Adiag =(
1 0
0 1
)
and the PV and PPV-Galois groups of partx Y = AdiagY are Gm and Gm(C ) respectively see [14
Proposition 39(2)] Therefore
ord(GRu(G)
)= ord(Gm(C )
)= 1
The matrix of M oplusP 1(Mdiag
)with respect to the appropriate basis is
1 tx + 1
x+1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Reductive LDAGs and the Galois Groups of Parameterized Linear Differential Equations 55
which is not completely reducible by (66) Therefore its PV group is not isomorphic to Gm the
PV group of Mdiag Thus G is not reductive In fact G is calculated in [41 Ex 34] yielding
G =(
e f
0 e
)isin Gm(C )timesGa(C )
∣∣∣∣∣ partt e = 0 part2t f = 0
Example 64 Consider the equation
part2x (y)+2xtpartx (y)+ t y = 0 (67)
The PPV-Galois group of this equation lies in GL2 One can make a standard substitution ([57
Exc 1355]) resulting in a new equation having PPV-Galois group in SL2 Once we know the PPV-
Galois group of this new equation results of [1] allow us to construct the PPV-group of the original
equation In our example the appropriate substitution is y = zeminusint