Reductive Linear Differential Algebraic Groups and the ...qcpages.qc.cuny.edu/~aovchinnikov/papers/computereductive.pdf · and realizes differential algebraic groups as Galois groups

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

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

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

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

= 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

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

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

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

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

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

Proof Let `∆ Grm rarr Gr m

a be the homomorphism

`∆(y1 yr ) =(part1 y1

part1 yr

yrpart2 y1

part2 yr

partm y1

partm 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

))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 )

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

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 =

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

)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

)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

which is a LAG Define

A0 =ϕ(K[GL(W )]) = K[H ] (32)

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

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

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)

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

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

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

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

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

)sub B n n isinN

S(xh) = S(xh minusx +x) = S(xh minusx)+S(x) isin (Bnminus1 + I )capB n

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

ϕ(xh) =ϕ(xh minusx +x) =ϕ(xh minusx)+ϕ(x) =ϕ(xh minusx)+0 isin Anminus1

I sub Kerα

On the other hand let α(x) = 0 Then there exists n isin N such that for all i 0 Eacute i Eacute n if xi isin B i

satisfy β(x) = x0 + +xn then

ϕ(xi ) isin Aiminus1

which implies that there exists yi isin I capBi such that

xi minus yi isin Biminus1

Therefore βminus1(xi ) isin I implying that

Kerαsub I

Thus α induces a Hopf algebra structure on gr A (In general if A is a filtered Hopf algebra then

gr A can be given (in a natural way) a structure of a graded Hopf algebra see eg [55 Chapter 11])

Consider the identity map (This map is differential if and only if K is constant)

γ KB rarr KB

of Hopf algebras Since

KB n)= KB n

J = γminus1(I)

is a Hopf ideal of KB Moreover it is differential since

partl(xh

)= (partl x

)h x isin KB

Therefore gr A has a structure of a differential Hopf algebra over K Furthermore it is differentially

generated by the Hopf algebra A0 sub gr A In other words gr A is isomorphic to the coordinate

algebra of an LDAG G (over K) dense in H By Corollary 37 G is connected Hence gr A has no

zero divisors

32 Subalgebras generated by W -filtrations

For n isinN let A(n) sub A denote the subalgebra generated by An Since An is a subcoalgebra of A it

follows that A(n) is a Hopf subalgebra of A Note that

A(n) n isinNforms a filtration of the vector

space A We will prove the result analogous to Proposition 38

Proposition 39 Suppose that G is connected If x isin A(n) y isin A(n+1) and x y isin A(n) then

y isin A(n)

Proof Let Gn n isin N stand for the LAG with the (finitely generated) Hopf algebra A(n) Since

A(n) sub A and A is an integral domain A(n) is an integral domain Let Gn+1 rarr Gn be the

epimorphism of LAGs that corresponds to the embedding A(n) sub A(n+1) and K be its kernel Then

we have

A(n) = AK(n+1)

Denote A(n+1) by B We have

x isin B K y isin B and x y isin B K

Let us consider this relation in QuotB sup B We have

y isin (QuotB)K capB = B K

Thus y isin A(n)

For s t isinN set

Ast = As cap A(t )

Since An sub A(n) Ast = As if s Eacute t Also As0 = A0 for all s isinZ+ Therefore one may think of Ast as

a filtration of the G-module V where the indices are ordered by the following pattern

(00) = 0 lt (11) = 1 lt (21) lt (22) = 2 lt (31) lt (32) lt (316)

(Note that t = 0 implies s = 0) We also have

As1t1 As2t2 sub As1+s2maxt1t2 (317)

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

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

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)

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

g 7rarr gminus1j g g j Since a conjugation preserves the U -filtration of B we conclude

g j (In) = (g j I

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)

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

]= 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

b2partl (x) = b(partl (bx)minusxpartl (b)) = bpartl (bx)minus (bx)partl (b) isin Bs+1t+1

partl (x) isin As+1t+1

We have

Bst sub Ast sub(

Ast)Γ

We will show that

Ast =(

Ast)Γ (323)

This will complete the proof as follows Suppose that

x isin B cap Ast sub(

Ast)Γ

By (323) there exists b isin B0 such that bx isin Bst Then Theorem 310 implies x isin Bst We

conclude (321)

Now let us prove (323) by induction on s the case s = 0 being already considered above

Suppose s Ecirc 1 Since Γ is a finite normal subgroup of the connected group G it is commutative

[5 Lemma V221] Therefore every Γ-module has a basis consisting of semi-invariant vectors

that is spanning Γ-invariant K-lines Therefore since a finite subset of the algebra A0 belongs to

a finite-dimensional subcomodule and A0 is finitely generated one can choose Γ-semi-invariant

generators X = x1 xr sub A0 of A Note that X differentially generates A Since Γ is finite its

scalar action is given by algebraic numbers which are constant with respect to the derivations of

K Hence the actions of Γ andΘ on A commute and an arbitrary product of elements of the form

θxi θ isinΘ is Γ-semi-invariant

Let 0 6= x isin (Ast )Γ We will show that x isin Ast Since a sum of Γ-semi-invariant elements is

invariant if and only if each of them is invariant it suffices to consider the case

x = prodjisinJ

θ j y j θ j isinΘ (324)

where J is a finite set and y j isin X sub A0 Moreover by Theorem 310 (324) can be rewritten to

satisfy sumjisinJ

ordθ j Eacute s and maxjisinJ

ordθ j

Eacute t

Since y j and θ j y j have the same Γ-weights

y = prodjisinJ

y j isin (A0)Γ = B0

Set g = |Γ| We have

y gminus1x = prodjisinJ

y gminus1j θ j (y j ) isin (

Ast)Γ

and for every j isin J

Aordθ j

If ordθ j lt s for all j isin J then by induction

y gminus1j θ j (y j ) isin Aordθ j ordθ j

for all j isin J This implies

y gminus1x isin Ast

Hence x isin Ast

Suppose that there is a j isin J such that ordθ j = s Let us set θ = θ j Then there exist i

1 Eacute i Eacute r and a isin A0 such that

x = aθ(xi ) isin AΓs

It follows that

axi isin AΓ0 = B0

We will show that x isin Ass = As There exist l 1 Eacute l Eacute m and θ isinΘ ord θ = s minus1 such that

θ = partl θ

If s = 1 then θ = partl and

xgi x = (axi )

(xgminus1

i partl xi)= (axi )partl

)g isin B1 sub A1

since xgi isin B0 Therefore x isin A1 Suppose that s Ecirc 2 We have

x = partl(aθ(xi )

)minuspartl (a)θ(xi )

Since u = aθ(xi ) isin (Asminus1)Γ by induction u isin Asminus1 Hence

partl (u) isin As

Since s Ecirc 2 we have

1 = ordpartl lt s and ord θ lt s

v = partl (a)θ(xi ) = x minuspartl (u) isin AΓs

by the above argument (for dealing with the case ordθ j lt s for all j isin J ) v isin As Therefore

x = partl (u)minus v isin As

4 Filtrations of G-modules in reductive case

In this section we show our main result the bounds for differential representations of semisim-

ple LDAGs (Theorem 45) and reductive LDAGs with τ(Z (G)) Eacute 0 (Theorem 49 note that

Lemma 214 implies that if K is differentially closed then a reductive DFGG has this property)

In particular we show that if G is a semisimple LDAG W is a faithful semisimple G-module and

V isin RepG then the W -filtration of V coincides with its socle filtration

41 Socle of a G-module

Let G be an LDAG Given a G-module V its socle socV is the sum of all simple submodules of V

The ascending filtration socn V nisinN on V is defined by

socn V

socnminus1 V = soc(V

socnminus1 V

) where soc0 V = 0 and soc1 V = socV

Proposition 41 Let n isinN

1 If ϕ V rarrW is a homomorphism of G-modules then

ϕ(socn V ) sub socn W (41)

2 If U V subW are G-modules and W =U +V then

socn W = socn U + socn V (42)

3 If V isin RepG then

socn (P i1

1 middot middotP imm (V )

)sub P i11 middot middotP im

socn V) (43)

Proof Let ϕ V rarrW be a homomorphism of G-modules Since the image of a simple module is

simple

ϕ(socV ) sub socW

Suppose by induction that

socnminus1 V)sub socnminus1 W

Set V =V

socnminus1 V W =W

socnminus1 W We have the commutative diagram

Vϕminusminusminusminusminusrarr WyπV

Vϕminusminusminusminusminusrarr W

where πV and πW are the quotient maps Hence

socn V)subπminus1

W ϕπV(

socn V)=πminus1

socV)subπminus1

W socW = socn W

where we used ϕ(

socV) sub socW Let us prove (42) Let U V sub W be G-modules It follows

immediately from the definition of the socle that

soc(U +V ) = socU + socV

Note that by (41) V cap socn W = socn V We have

W socn W = (U

socn W

socn U

socn V

Applying soc we obtain statement (42)

In order to prove (43) it suffices to do it only for Pi (V ) since the other cases would follow

by induction Let

πi Pi (V ) rarrV

be the natural epimorphism from (22) We have πminus1i (U ) = Pi (U )+V for all submodules U sub V

Hence by (41)

socn Pi (V ) subπminus1i

(socn V

)= Pi(

socn V)+V

Since socn socn M = socn M for an arbitrary module M

socn Pi (V ) = socn socn Pi (V ) sub socn (Pi

(socn V

)+V)sub Pi

(socn V

)+ socn V = Pi(

socn V)

Proposition 42 Suppose that

soc(U otimesV ) = (socU )otimes (socV )

for all U V isin RepG Then

socn(U otimesV ) =nsum

(soci U

)otimes (socn+1minusi V

for all U V isin RepG and n isinN

Proof For a G-module V denote socn V by V n n isinN Suppose by induction that (44) holds for

all n Eacute p and U V isin RepG Set

Sp = Sp (U V ) =psum

i=1U i otimesV p+1minusi

For all 1 Eacute i Eacute p we have

Fi = (U i otimesV p+2minusi )(

Sp cap (U i otimesV p+2minusi ))= (

U i otimesV p+2minusi )(U iminus1 otimesV p+2minusi +U i otimesV p+1minusi )

Fi (U i U iminus1)otimes (

V p+2minusi V p+1minusi )

By the hypothesis Fi is semisimple Hence so is

Sp+1Sp =psum

i=1Fi sub (U otimesV )Sp

By the inductive hypothesis we conclude

socp+1(U otimesV ) sup Sp+1

Now we prove the other inclusion Let

ψ U rarr U =U U 1

be the quotient map Note the commutative diagram

U otimesVπminusminusminusminusminusrarr X = (U otimesV )

SpyψotimesId

yU otimesV

πminusminusminusminusminusrarr X = (U otimesV

)Spminus1

where π and π are the quotient maps By the inductive hypothesis we have

socp+1(U otimesV ) =πminus1(X 1)sub (ψotimes Id)minus1(πminus1)(soc X)= (ψotimes Id)minus1(socp (

U otimesV))sub Sp+1

since ψminus1(

soci U)= soci+1 U

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

H diff

is associated with a grading (see proof of Proposition 38) In particular the sum I of

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)

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

socn+1(V otimesW ) =nsum

(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)

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

be the identity component of the center of G The LDAG S is semisimple and the multiplication

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

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

``(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

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 )

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

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

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

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

Furthermore if Z is a solution matrix of partY = AY then

(Z parti 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 )

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

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

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

differential equation in block upper triangular form

partY =

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

)rarrG sub GL(ω(M))

be the morphisms (of LDAGs) corresponding to a Levi decomposition of G Then ρdiagsim=micro

Proof Since ρdiag is completely reducible ω(Mdiag

)is a completely reducible ρdiag

)-module

Therefore ρdiag

)is a reductive LAG [54 Chapter 2] Hence

)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

) (58)

Since all Levi K part-subgroups of G are conjugate (by K part-points of Ru

)) [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

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

)capG is a

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

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

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)

(E) Assume that we are given F an algebraic extension of K(x) and A isin Mn(F ) and B1 B` isin F n

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]

(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

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

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)

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

[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])

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

532 An algorithm to compute the maximal reductive quotient GRu(G) of a PPV-Galois group

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

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

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

) 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

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

Hi = H1 middot middotHiminus1 middotHi+1 middot middotH`

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

)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

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

(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

) 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

diag Hdiag

](as in (A)) Using the solution

to (D) one calculates a differential equation partY = BY whose PV-Galois group is

Hdiag H

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

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

11x22 minusx prime12x21

A = Kx11 x12 x21 x22

[x11x22 minusx12x21 minus1] (61)

which induces the following differential representation of SL2

SL2(U ) 3(

)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

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

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)

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

1 tx + 1

whose PV-group is (a b

) ∣∣∣ 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 =(

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 )

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

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

)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

xt We find that z satisfies the

equation

part2x (y)minus (

14(2xt )2 + (2xt )prime2minus t)y = 0 lArrrArr part2

x (y)minus (xt )2 y = 0 (68)

which now has PPV-Galois group in SL2 and eminusint

xt satisfies the equation

partx (y)+ ((2xt )2)y = 0 lArrrArr partx (y)+ (xt )y = 0 (69)

which has PPV-Galois group in GL1 = Gm We shall refer to Equations (68) and (69) as the

auxiliary equations A calculation on MAPLE using the kovacicsols procedure of the DEtools

package shows that the PV Galois group H of (68) is SL2 Since for all 0 6= n isin Z U (x) has no

solutions of

partx (y)+ (nxt )y = 0

the PV Galois group of (69) is Gm Therefore by [1 Section 34] the PV Galois group of (67) is

GL2sim= (SL2timesGm)

1minus1

Hence the PPV-Galois group G of (67) is of the form

G = (G1 timesG2)

1minus1 sub H

where G2 is Zariski-dense in Gm and G1 is conjugate in GL2 either to SL2 or SL2(C ) We will now

calculate G1 and G2 For the former note that the matrix form of (68) is

partx Y =(

(xt )2 0

Since for the matrix

which can be found using the dsolve procedure of MAPLE one has partx (B)minuspartt (A) = [AB ] (68)

is completely integrable and therefore G1 is conjugate to SL2(C ) To find G2 compute the first

prolongation of (69)

A1 =(minusxt minusx

0 minusxt

Setting

1 1minus2x2

minus2x2+t

we see that

Cminus1 A1C minusCminus1partx (C ) =(

2minusx2tx 0

0 x(2minusx2tminust 2)x2+t

Hence the differential equation corresponding to A1 is completely reducible Therefore G2 =Gm(C ) that is

G sim= GL2(C )

Note that C can be found using the dsolve procedure of MAPLE

Example 65 Starting with

part2x (y)minus 2t

xpartx (y) = 0 (610)

the auxiliary equations will be

part2x (y)minus t (t +1)

x2 y = 0 and partx (y) = t

The PPV-Galois group of the latter equation is

G2 = g isin Gm | (part2

t g )g minus (partt g )2 = 0

For the former equation a calculation using dsolve from MAPLE shows that there is no B isinM2(U (x)) such that partx (B)minuspartt (A) = [AB ] where

0 1t (t+1)

which implies that this equation is not completely integrable Therefore G1 = SL2 Thus the PPV-

Galois group of (610) is

g isin GL2 | (part2

t det(g ))

det(g )minus (partt det(g ))2 = 0

Funding

AM was supported by the ISF grant 75612 AO was supported by the NSF grant CCF-0952591

MFS was supported by the NSF grant CCF-1017217

References

[1] C Arreche Computing the differential Galois group of a one-parameter family of second order linear

differential equations 2012 URL httparxivorgabs12082226

[2] C Arreche A Galois-theoretic proof of the differential transcendence of the incomplete Gamma

function Journal of Algebra 389119ndash127 2013 URL httpdxdoiorg101016jjalgebra

201304037

[3] M A Barkatou On rational solutions of systems of linear differential equations Journal of Sym-

bolic Computation 28(4-5)547ndash567 1999 URL httpdxdoiorgproxlibncsuedu101006

jsco19990314

[4] A Beilinson V Ginzburg and W Soergel Koszul duality patterns in representation theory Jour-

nal of the American Mathematical Society 9473ndash527 1996 URL httpdxdoiorg101090

S0894-0347-96-00192-0

[5] A Borel Linear Algebraic Groups Springer 2nd enlarged edition 1991 URL httpdxdoiorg10

1007978-1-4612-0941-6

[6] N Bourbaki Algegravebre Chapitre 8 Modules et anneaux semi-simples Springer 2012 URL http

dxdoiorg101007978-3-540-35316-4

[7] A Braverman P Etingof and D Gaitsgory Quantum integrable systems and differential Galois theory

Transformation Groups 2(1)31ndash56 1997 URL httpdxdoiorg101007BF01234630

[8] P Cassidy Differential algebraic groups American Journal of Mathematics 94891ndash954 1972 URL

httpwwwjstororgstable2373764

[9] P Cassidy The differential rational representation algebra on a linear differential algebraic group

Journal of Algebra 37(2)223ndash238 1975 URL httpdxdoiorg1010160021-8693(75)90075-7

[10] P Cassidy Unipotent differential algebraic groups In Contributions to algebra Collection of papers

dedicated to Ellis Kolchin pages 83ndash115 Academic Press 1977

[11] P Cassidy Differential algebraic Lie algebras Transactions of the American Mathematical Society 247

247ndash273 1979 URL httpdxdoiorg1023071998783

[12] P Cassidy Differential algebraic group structures on the plane Proceedings of the American

Mathematical Society 80(2)210ndash214 1980 URL httpdxdoiorg1023072042948

[13] P Cassidy The classification of the semisimple differential algebraic groups and linear semisimple

differential algebraic Lie algebras Journal of Algebra 121(1)169ndash238 1989 URL httpdxdoi

org1010160021-8693(89)90092-6

[14] P Cassidy and M Singer Galois theory of parametrized differential equations and linear differential

algebraic group IRMA Lectures in Mathematics and Theoretical Physics 9113ndash157 2007 URL http

dxdoiorg104171020-17

[15] E Compoint and M Singer Computing Galois groups of completely reducible differential equations

Journal of Symbolic Computation 28(4-5)473ndash494 1999 URL httpdxdoiorg101006jsco

19990311

[16] W A de Graaf Lie algebras theory and algorithms volume 56 of North-Holland Mathematical Library

North-Holland Publishing Co Amsterdam 2000 ISBN 0-444-50116-9 URL httpdxdoiorg

proxlibncsuedu101016S0924-6509(00)80040-9

[17] P Deligne and J Milne Tannakian categories In Hodge cycles motives and Shimura varieties

volume 900 of Lecture Notes in Mathematics pages 101ndash228 Springer-Verlag Berlin 1981 URL

httpdxdoiorg101007978-3-540-38955-2_4

[18] T Dreyfus Computing the Galois group of some parameterized linear differential equation of order

two To appear in Proceedings of the American Mathematical Society 2014 URL httparxivorg

abs11101053

[19] T Dreyfus A density theorem for parameterized differential Galois theory To appear in the Pacific

Journal of Mathematics 2014 URL httparxivorgabs12032904

[20] D Eisenbud C Huneke and W Vasconcelos Direct methods for primary decomposition Inven-

tiones Mathematicae 110(2)207ndash235 1992 URL httpdxdoiorgproxlibncsuedu10

1007BF01231331

[21] H Gillet S Gorchinskiy and A Ovchinnikov Parameterized PicardndashVessiot extensions and Atiyah

extensions Advances in Mathematics 238322ndash411 2013 URL httpdxdoiorg101016jaim

201302006

[22] S Gorchinskiy and A Ovchinnikov Isomonodromic differential equations and differential categories

Journal de Matheacutematiques Pures et Appliqueacutees 2014 URL httpdxdoiorg101016jmatpur

201311001 In print

[23] D Y Grigoriev Complexity for irreducibility testing for a system of linear ordinary differential

equations In M Nagata and S Watanabe editors Proceedings of the International Symposium on

Symbolic and Algebraic Computation- ISSACrsquo90 pages 225ndash230 ACM Press 1990 URL httpdx

doiorg1011459687796932

[24] C Hardouin and M Singer Differential Galois theory of linear difference equations Mathematische

Annalen 342(2)333ndash377 2008 URL httpdxdoiorg101007s00208-008-0238-z

[25] G P Hochschild Basic Theory of Algebraic Groups and Lie Algebras Springer New York 1981 URL

httpdxdoiorg101007978-1-4613-8114-3

[26] E Hrushovski Computing the Galois group of a linear differential equation Banach Center

Publications 5897ndash138 2002 URL httpdxdoiorg104064bc58-0-9

[27] J E Humphreys Linear algebraic groups Springer-Verlag New York 1975 URL httpdxdoiorg

101007978-1-4684-9443-3 Graduate Texts in Mathematics No 21

[28] M Kamensky Tannakian formalism over fields with operators International Mathematics Research

Notices 2013(24)5571ndash5622 2013 URL httpdxdoiorg101093imrnrns190

[29] M Kamensky Model theory and the Tannakian formalism To appear in the Transactions of the

American Mathematical Society 2014 URL httparxivorgabs09080604

[30] I Kaplansky An Introduction to Differential Algebra Hermann Paris 1957

[31] R Kodera and K Naoi Loewy series of Weyl modules and the Poincareacute polynomials of quiver

varieties Publications of the Research Institute for Mathematical Sciences 48(3)477ndash500 2012 URL

httpdxdoiorg102977PRIMS77

[32] E Kolchin Algebraic matric groups and the PicardndashVessiot theory of homogeneous linear ordinary

differential equations Annals of Mathematics 49(1)1ndash42 1948 URL httpwwwjstororg

stable1969111

[33] E Kolchin Differential Algebra and Algebraic Groups Academic Press New York 1973

[34] E Kolchin Differential Algebraic Groups Academic Press New York 1985

[35] J Kovacic An algorithm for solving second order linear homogeneous differential equations Jour-

nal of Symbolic Computation 2(1)3ndash43 1986 URL httpdxdoiorg101016S0747-7171(86)

80010-4

[36] P Landesman Generalized differential Galois theory Transactions of the American Mathematical

Society 360(8)4441ndash4495 2008 URL httpdxdoiorg101090S0002-9947-08-04586-8

[37] O Leoacuten Saacutenchez Geometric axioms for differentially closed fields with several commuting derivations

Journal of Algebra 362107ndash116 2012 URL httpdxdoiorg101016jjalgebra201203

[38] W Magnus A Karrass and D Solitar Combinatorial group theory Presentations of groups in terms of

generators and relations Interscience Publishers [John Wiley amp Sons Inc] New York-London-Sydney

[39] A Minchenko and A Ovchinnikov Zariski closures of reductive linear differential algebraic groups

Advances in Mathematics 227(3)1195ndash1224 2011 URL httpdxdoiorg101016jaim2011

[40] A Minchenko and A Ovchinnikov Extensions of differential representations of SL2 and tori Journal

of the Institute of Mathematics of Jussieu 12(1)199ndash224 2013 URL httpdxdoiorg101017

S1474748012000692

[41] A Minchenko A Ovchinnikov and M F Singer Unipotent differential algebraic groups as parame-

terized differential Galois groups To appear in the Journal of the Institute of Mathematics of Jussieu

2014 URL httpdxdoiorg101017S1474748013000200

[42] C Mitschi and M Singer Monodromy groups of parameterized linear differential equations with

regular singularities Bulletin of the London Mathematical Society 44(5)913ndash930 2012 URL http

dxdoiorg101112blmsbds021

[43] C Mitschi and M Singer Projective isomonodromy and Galois groups Proceedings of

the American Mathematical Society 141(2)605ndash617 2013 URL httpdxdoiorg101090

S0002-9939-2012-11499-6

[44] A Ovchinnikov Tannakian approach to linear differential algebraic groups Transformation Groups

13(2)413ndash446 2008 URL httpdxdoiorg101007s00031-008-9010-4

[45] A Ovchinnikov Tannakian categories linear differential algebraic groups and parametrized linear

differential equations Transformation Groups 14(1)195ndash223 2009 URL httpdxdoiorg10

1007s00031-008-9042-9

[46] R Ree Commutators in semi-simple algebraic groups Proceedings of the American Mathematical

Society 15457ndash460 1964 URL httpdxdoiorg101090S0002-9939-1964-0161944-X

[47] M Singer Algebraic solutions of nth order linear differential equations In Proceedings of the Queenrsquos

University 1979 Conference on Number Theory volume 54 of Queenrsquos Papers in Pure and Applied

Mathematics pages 379ndash420 Queenrsquos University Kingston ON Canada 1980

[48] M Singer Liouvillian solutions of nth order homogeneous linear differential equations American

Journal of Mathematics 103(4)661ndash682 1981 URL httpdxdoiorg1023072374045

[49] M Singer Liouvillian solutions of linear differential equations with Liouvillian coefficients Journal

of Symbolic Computation 11(3)251ndash273 1991 URL httpdxdoiorg101016S0747-7171(08)

80048-X

[50] M Singer Linear algebraic groups as parameterized PicardndashVessiot Galois groups Journal of Algebra

373(1)153ndash161 2013 URL httpdxdoiorg101016jjalgebra201209037

[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

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