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United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THIRD HOMOLOGY GROUPSOF UNIVERSAL CENTRAL EXTENSIONS OF A LIE ALGEBRA

Allahtan Victor Gnedbaye*Universite de N'Djamena, Faculte des Sciences Exactes et Appliquees,

Departement de Mathematique et d'Informatique, B.P. 1027, N'Djamena (TCHAD)and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Leibniz algebras are a non-commutative generalization of usual Lie algebras. We studyuniversal central extensions of a perfect Leibniz algebra C and we prove that the kernelis isomorphic to the second homology group HL2(£). Next, given a perfect Lie (henceLeibniz) algebra g, we compute and compare the homology groups HL3(il), HL3(u) andH3(u), where i! (resp. u) is the universal central extension of g, in the category of Leibniz(resp. Lie) algebras, and where HL* (resp. H*) stands for the Leibniz (resp. the classicalChevalley-Eilenberg) homology theory.

MIRAMARE - TRIESTE

June 1998

E-mail: [email protected] (until January 1999).

Typeset by .4^(5-

0. Introduction

Discovered by Jean-Louis Loday (see [7]), Leibniz algebras are a non-commutative varia-tion of usual Lie algebras. There is an homology theory HL* for these new algebraic objects(see [8]), whose properties are similar to those of the classical Chevalley-Eilenberg homologytheory H* for Lie algebras (see [2]).

In this paper, we study universal central extensions of a Leibniz algebra. Mimicking anarticle by H. Garland (see [3]), we give a criterion for a central extension to be universal.Then we deduce a criterion for a Leibniz algebra to admit a universal central extension(perfectness). We show that the kernel of the universal central extension is canonically iso-morphic to the second homology group of the initial object. Since Lie algebras are examplesof Leibniz algebras, any perfect Lie algebra g admits a universal central extension il (resp.u) in the category (Leib) (resp. (Lie)) of Leibniz (resp. Lie) algebras. It turns out thatthe Leibniz algebra il is the universal central extension of u in the category (Leib), andthat u = iine that is, the Lie algebra canonically associated to il. These universal centralextensions are homologically characterized by the following isomorphisms

HLi(il) = HL2(il) = 0 = Hi(u) = H2(u),

ker(HL2(fl) -^>H2(fl)) ^HL 2 (u) =* ker(il -» u =* i lL i e) .

Next, we compute the homology groups HL3(il), HL3(u) and H3(u) in terms of the homol-ogy groups HL*(g) and H*(g). This is done by using the Hochschild-Serre and its Leibnizversion spectral sequences (see [4], [5]). We give an interpretation of the natural mapsHL3(il) -> HL3(u) and HL3(u) -^ H3(u) as follows

Theorem. Let il {resp. u) be the universal central extension in the category (Leib) {resp.(Lie)) of a perfect Lie algebra g. Then we have a {non-natural) commutative diagram

HL3(il) -

IHL3(u) -

I

> HL^g)1^

I— = - • HL2(fl)®2/#®2

I

©

©

HL3(fl)

1HL3(fl)

IH3(u) — ^ S2(H2(fl)) © H3(fl)

whereK := ker(HL2 (g) -» H2 (g)) ^ ker(il -^ u ^ iiLie) ^ HL2 (u)

and S is £/ie symmetric functor.

The symbol K denotes a fixed commutative ring with unit. All modules, linear maps andtensor products involved here are over K. In order to simplify computations, we will assumethat K is a field in the section 3.

1. Prerequisites on Leibniz algebras

1.1. Leibniz algebras. A Leibniz algebra is a K-module £ equipped with a bilinear map[—,—]:£ X £ —7- £, called bracket and satisfying only the Leibniz identity

[ x , [ y , z ] ] = [ [ x , y ] , z ] - [[x,z],y]

for any x,y,z £ £• In the presence of the condition [x,x] = 0, the Leibniz identity isequivalent to the so-called Jacobi identity] therefore Lie algebras are examples of Leibnizalgebras.

A morphism of Leibniz algebras is a linear map / : L\ —> £2 such that

for any x,y £ L\. It is clear that Leibniz algebras and their morphisms form a categorythat we denote by (Leib).

A two-sided ideal of a Leibniz algebra £ is a submodule % such that [x,y] £ % and[y, x] £ % for any x £ % and any y £ £. For any two-sided ideal % in £, the quotient moduleCjH inherits a structure of Leibniz algebra induced by the bracket of £. In particular, let([a;, a;]) denotes the two-sided ideal in £ generated by all brackets [a;, a;]; then the Leibnizalgebra C/([x, x]) is in fact a Lie algebra, said canonically associated to £ and is denoted by^Lie-

Let £ be a Leibniz algebra. Denote by £ ' := [£,£] the submodule generated by allbrackets [x,y]. The Leibniz algebra £ is said to be perfect when £.' = £.. It is clear that anysubmodule of £ containing £ ' is a two-sided ideal in £.

1.2. Non-trivial examples, i) If (g, [—,—], d) is a differential Lie algebra, then thebracket defined by [s,y]d := [x,d(y)] satisfies the Leibniz identity (but obviously it is notskew-symmetric).

ii) Let M be a representation of a Lie algebra g (the action of g on M being denoted bymx for TO £ M and x £ g). For any g-equivariant map / : M —> g, the bracket given by[TO, TO'] := m-f('m ' induces a structure of Leibniz (non-Lie) algebra on M.

1.3. Semi-representations. Let £ be a Leibniz algebra. A semi-representation of £ is aK-module M equipped with an action of £, [—, — ] : M X £ —> M, satisfying the rule

[ T O , [X, y]] = [ [ T O , x ] , y ] - [ [ T O , y], x]

for any TO £ M and any x,y £ £. It turns out that a semi-representation of a Leibnizalgebra £ is equivalent to a representation the Lie algebra C-Lie in the classical sense. It isclear that a Leibniz algebra is a semi-representation over itself by the adjoint action.

In [10], there are notions of (co) representations with a suitable notion of universal en-veloping algebra of a Leibniz algebra.

1.4. Leibniz h o m o l o g y . Let £ be a Leibniz algebra and let M be a semi-representationof £ . There is a well-defined complex (T*(£ , M) := M <8> £ ® *, d) where the boundary mapd : T n ( £ , M) -^ T1 1"1 (£ , M) is given by the formula (see [7])

d(x0,--- ,xn) := Y^ (-1)1+1(xo,--- , a:;_i ,[a:; ,a:j] ,a:;+ i ,--- , £ ) , • • • ,xn)

for any x0 £ M and x\, • • • , xn £ £; here (x0, • • • , xn) stands for xo(x) • • • (x)xn E Tn(£, M).The homology of this complex is denoted HL* (£, M), and simply HL* (£) if M = K equippedwith the trivial action of £. One easily checks that

HL0(£) = K, EU(£)^£ab :=£/[£,£],ELO(£,M)^MC:=M/[M,£],

Remark that if g is a Lie algebra, then the complex (T*(fl, M), d) is nothing but a liftingof the classical Chevalley-Eilenberg complex (M ® A* (g), d) defining the homology H*(g,Af)of Lie algebras (see [2]). Moreover the canonical projection can* : M ® £® * —> M ® A*(£)is a morphism of complexes which induces in homology isomorphisms in degrees 0 and 1,and an epimorphism in degree 2.

Furthermore, J.-L. Loday and T. Pirashvili define general (co)homology theories with co-efficients in (co)representations, and they give a Tor-Ext interpretation for Leibniz (co)homology.

2. Universal central extensions of a Leibniz algebra

2.1. Central extensions. An exact sequence of Leibniz algebras

is called extension of £ by H. The morphism i is a Leibniz algebra isomorphism from Honto the kernel ker(p) of the extension. Therefore we merely write p : £ -» £ the extension(£), and by abuse of language, we say that the Leibniz algebra £ is an extension of £.

An extension p : £ -» £ is said to be split in the category (Leib) if there exists a Leibnizalgebra morphism s : £ —> £ such that ps = id/;; the map s is called a section of p.

A central extension of a Leibniz algebra £ is an extension p : C -» £ whose kernel satisfies[ker(p),C] = [C,ker(p)] = O.

A central extension a : li -» £ is said to be universal if for any central extension p : C -» £there exists a unique Leibniz algebra morphism <j> :IA —> C such that p<j> = a. It is clear thata universal central extension, when it exists, is unique (up to a unique isomorphism).

2.2. Universality criterion. Here are some properties which characterize the universalityof a central extension.

Proposition 2.1. If a central extension a :U —» £ is universal, then the Leibniz algebra Uis perfect.

Proof. Assume that the Leibniz algebra U is not perfect; then there is a non-trivial map/ :U/[U,U] -^ K.

Equip the direct sum U © K with the Leibniz algebra structure given by

where x, x' £ li and A, A' £ K. Clearly the map a. : li © K —> £, (x, A) H-> a{x) is a surjectivemorphism of Leibniz algebras whose kernel (which is nothing but ker(a) © K) is central inIA © K. Thus the extension a : ZY © K ^> £ is central.

O n e c h e c k s t h a t t h e m a p s (f>i,(f>2 ' . l A ^ - l A ® ^ g i v e n b y 4>\{x) : = ( x , 0 ) a n d ^ ( x ) '•=(x,f(x)), are two distinct morphisms of Leibniz algebras such that axfri = a, (i = 1,2);which contradicts the universality of the extension IA. •

Proposition 2.2. If a central extension a :14 -» £ is universal, then any central extensionofU splits in the category (Leib).

Proof. This is done in two steps.i) Let p : £ —» IA be a central extension such that £ is perfect. It is clear that the map

/3 := ap : £ —> £ is a surjective morphism of Leibniz algebras. Let us show that its kernel iscentral in £. Firstly remark that an element z is in ker(/3) if, and only if, p(z) is in ker(a).Since ker(a) is central in IA, if x or y is in ker(/3), then [x,y] is in ker(p). Since ker(p) iscentral in £, one has [[x, y], z] = 0 (resp. [x, [y, z]] = 0) if x or y (resp. y or z) is in ker(/3).Consequently, by the Leibniz identity, we get

[x, [2/1,2/2]] = [[x, 2/1], 2/2] - [[z, 2/2], 2/i] = 0,

[[2/1,2/2], z] = [2/1, [2/2, z]] + [[2/1, z], 2/2] = 0

for any a; G ker(/3), 2/1,2/2 £ £. Therefore ker(/3) is central in [£,£] = £.By the universality of the extension a : IA -» £, there exists a Leibniz algebra morphism

s :IA —> £ such that /3s = a i.e., aps = a. Consequently the morphism ip := ps — id^ takesvalue in ker(a). Since ker(a) is central in IA, one has

i>([x,y\) = \ps(x),ps(y)] - [x,y] = [ip(x),ps(y)]+ [x,ip(y)] = 0.

It follows that if) is trivial on \U,U\, from whence ps = id^ since li = \U,W\ (cf. Proposition2.1).

ii) Let p : £ -» li be any central extension. Denote by p' the restriction of p to thesubalgebra £' = [£,£]. Since IA is perfect, it is clear that the morphism p' : £' —> IA is stillsurjective with a central kernel. Let us show that £' is perfect. Let [x,y] be a generator of£' . Since IA is perfect and p is surjective, one can successively write

Therefore the element h := x — ̂ [xi, x[] is in ker(p). By the same way, there exist elements(j/j, y'j) in £ such that h' := y — ^ [Vji Uj] 1S m ker(p). Since ker(p) is central in £, we get

Then the subalgebra £' is perfect. And by the step i) there exists an algebra morphisms' : IA —7- £' splitting p'. Denoting by 1 the inclusion map £' M- £, the composed maps := is' : ZY —> £ is an algebra morphism splitting p. •

Now we can state the following

Theorem 2.3. Let £ be a Leibniz algebra. A central extension IA of £ is universal if, andonly if, the Leibniz algebra IA is perfect and any central extension ofIA splits.

Proof of the converse. Let p : C -» £ be a central extension of £. We have to showthat there is a unique algebra morphism (f> : IA —> C such that pcj) = a. The uniquenessfollows from

Lemma 2.4. If a :IA -» £ is a central extension and if the Leibniz algebra IA is perfect, thenfor any central extension p : C —» £ there exists at most one algebra morphism (f> : IA —> Csuch that p<f) = a.

Proof of the Lemma. In fact, suppose that there are two such morphisms <j> and <j>'.Then for all x,y G IA one has

{4> - 4>'){[x, y}) = [4>{x),

Since the map <j> — <j>' takes value in ker(p) which is central in C, it follows that <j> — <j>' istrivial on [14,14] = 14. From whence follows the Lemma. •

For the existence of such a morphism, consider the product C X li equipped with thebracket given by [(c, u), (c', u')] := ([c, c'], [u, u']). Denote by C XcU the subalgebra

CxcK := {(c, u) eC xK \ p(c) = a(u)}.

The second projection p2 :CX£ZY—>ZYisa surjective morphism of algebras whose kernel isobviously central in C XcU. Therefore there exists a morphism s :IA —> C XcU such thatp2s = idij. Consider the morphism <j> := p\s : li —> C where p\ : C XcU —> C is the firstprojection. By definition of p\, p2, <j) a n ( i si o n e n a s S(M) = (Pis(u)i P2S(U)) = (<^(M))M)-But (4>(u), u) £C XcU means that p<j)(u) = a{u); from whence p<j> = a, and the Theorem isproved. •

2 . 3 . Remark . These properties of the universal central extension are homologically char-acterized by the equalities

HLi (£) = HLi(W) = HL 2 ( ^ ) = 0.

2.4. Existence criterion. Now we characterize Leibniz algebras which admit a universalcentral extension.

Theorem 2.5. A Leibniz algebra L {free as a M^-module) admits a universal central exten-sion if, and only if, it is perfect. Moreover, the kernel of the universal central extension iscanonically isomorphic to HL2 (£).

Proof. The condition is necessary because a universal central extension is perfect andthe surjective image of a perfect Leibniz algebra is also perfect.

Conversely, suppose that the Leibniz algebra C is perfect. Let lva(d3) be the image of theLeibniz boundary d3 i.e., the submodule of £® 2 generated by the elements

d 3 ( x ® y ® z ) = [ x , y ] ® z - [ x , z ] ® y - x ® [ y , z ] , V x , y , z G JC.

Consider the quotient module M := £®2/lva(d3), equipped with the trivial action of C.The canonical projection n : C®2 —> M is obviously a 2-cocycle of HL (£,Af). Then itdetermines a central extension Cv; recall that Cv is the K-module Cv = £ © M equippedwith the bracket given by [(x,m), {x',m')] := ([x, x'], n(x, x')) (see [10]). Let us show thatthe subalgebra L'v := [£„,£„] is perfect. First, remark that we have £v = L'v + M. Infact, since C is perfect, any element of Cv takes the form (J^ [x{, a;'], TO); by definition of thebracket on Cv, one has

^i, 0), ( 4 0)] + (0, TO -

which proves the equality Cv = C'v + M. Since M is central in Cv, one gets

C'v = [CV,CV] = [C'v + M,C'7T + M] = [£'„,£'„].

From whence follows the perfectness of the algebra C'v.

The first projection p\ : C'v —> C is a central extension whose kernel is generated by theelements of the form (0, ̂ ir(xi, a;')) such that ^ [xii x'i] = 0-

Since the K-module C is free, any central (hence abelian) extension of C is of the formp : JCJ -» JC for a well-determined 2-cocycle / of HL (C, V). Then the map

is an algebra morphism satisfying pcf> = p\. The uniqueness follows from the Lemma 2.4 (L'vis perfect).

Now let us show that ker(pi) is canonically isomorphic to HL2(£). Since the Leibnizboundary <i2 : C®2 —> £ acts by x® y \—> [x, y], the K-linear map

—>HL2(£), (0, J > ( ^ , x'J) ^ J ] xt ®x\

is well-defined and surjective. Suppose that £(0, ^ir(xi, x[)) is a boundary i.e.,

Then, by the definition of M = C®2 /lm(dz), one has

d^y^y'^y") = 0.

Thus the map £ is also injective; which proves the isomorphism ker(pi) = HL2(£). •

2.5. Remark . The universal central extension can also be characterized by the following.Consider HL2(£) as a trivial representation of C By the universal coefficient theorem, thereis an isomorphism HL (£,HL2(£)) = Hom(HL2(£), HL2(£)). But one knows that there is anatural bijection HL2(£, HL2(£)) ^ Ext(£, HL2(£)), where Ext(£, HL2(£)) denotes the setof isomorphism classes of abelian extensions of £ by HL2(£). The universal central extensioncorresponds to the element

idHL2(£) GHom(HL2(£),HL2(£)).

2.6. Case of Lie algebras. Let g be a perfect Lie algebra which is free as a K-module.Then there exist a universal central extension a : il ^> g in the category (Leib) of Leibnizalgebras, and a universal central extension a' : u ^> g in the category (Lie) of Lie algebras.

Proposition 2.6. / / the K-module u is free (e.g. when K is a field), then the Leibnizalgebra il is the universal central extension of the perfect Lie algebra u in the category(Leib). Moreover there is an isomorphism of Lie algebras u —> iine-

Proof. In fact, the extension a' : u ^> g is also central in the category (Leib). Thereforethere exists a morphism of Leibniz algebras (f> : il —> u such that a'(f> = a. Since il is perfectand any central extension il splits, it suffices to show that the map <j> : il —> u is a centralextension. Since ker(<̂ >) C ker(a), it is clear that ker(<̂ >) is central in il. Let us show that (f>is surjective. First, remark that one has u = Im((f>) + ker(a'). In fact, let z' £ u; since a issurjective and a'<f) = a, there exists an element z G il such that a'(z') = a(z) = a'(f)(z) i.e.,z' — (f>(z) G ker(a'). From whence the equality u = Im((f>) + ker(a'). Since the algebra u isperfect and ker(a') is central in u, we have

u = [u,u] = [lm((f>) + ker(a'),Im(^) + ker(a')] = [Im(^), Im(^)] C

Therefore the morphism <j> is surjective.

Furthermore, denote by a (resp. <f>) the morphism induced by a (resp. <f>) on iine- Itis clear that the extension a : Hue -» 9 ls central in the category (Lie) and that a'<f) = a.Therefore there exists a Lie algebra morphism ip : u —> iine such that ~aip = a'. Bycomposition, we get

a(ip<f)) = a'(f) = a and a'(<f)ip) = aip = a'.

Since the Lie algebras u and iine a r e perfect, we deduce from the Lemma 2.4 that <f)ip = idon u and ip(f> = id on il; from whence the isomorphism u —> iine- D

As consequence, we obtain a commutative diagram with exact columns and rows

0

I0 > ker(cara2) > HL2(u) > 0

I I I0 > HL2( f l) > U > s

I0 > H2(fl)

I I I0 0 0

(Leib)

which yields the following characterization

Corollary 2.7. With the above notation, one has the isomorphisms

ker(il >̂ iiLie) ^ HL2(ilLje) ^ ker(can2 : HL2(fl) >̂ H2(fl)). •

2.7. Examples: the Steinberg and Virasoro algebras. Let A be an associative algebraand let sln(A) be the Lie algebra of (n X ra)-matrices with entries in A and with zero tracein the abelianised A/[A, A] (here [A, A] stands for the submodule of A generated by thecommutators [a, b] = ah — ba, for any a, b £ A). Recall that the Lie algebra sln(A) is perfectfor n > 3.

Studying the universal central extension stn(A) of sln(A) in the category (Lie), Ch.Kassel and J.-L. Loday (see [6]) generalize a result by S. Bloch (see [1]) and obtain theisomorphism

H2(s[n(A))^HCi(A), Vn > 5

where HCi(A) denotes the cyclic homology of A (see [7]).

J.-L. Loday and T. Pirashvili (see [10]) construct the universal central extension st[n(A)of sln(A) in the category (Leib) and obtain the isomorphism

Vn>5

where HHi(A) denotes the Hochschild homology of A. Corollary 2.7 implies that

stn{A)^stln(A)Lte.

They also show that there is an isomorphism

EUiDeriCiz^-1])) ^ E2(Der(C[z, z-1]))

which proves that the two universal central extensions of the Lie algebra Der(C[z, z~1])coincide. In fact, this universal central extension is the Virasoro algebra.

3. Third homology groups

From now on we assume that K is a field.

3.1. Helpful background. While making explicit computations, one often needs thefollowing characterization of spectral sequences, whose cohomological version can be foundin [11, section 2.2.2].

Let (F£)p>o be a filtration of a complex (C*, d) and let {Er)r>i be the spectral sequencederived from this filtration. Define

Z;tq := F?+q n d-1 ( i * ^ ) and B^q := F?+q n d(Fgrq+1).

Then one hasrpr r~j yr i (yr — 1 , nr-h^VA ~ P,ql V^p-l.g + l ~r Dp,q )•

The differential maps are given by the commutative diagram

yr d y yrp,q p — r,q-\-r — l

v'q ) T?rp — r,q-\-r — l

where rj^q : Z'^q -» E'^q is the canonical projection with ke r (^ g ) = ZrpZ\A+1 + Br

p~ql

Moreover we have

q-i H d(Fp+q) = Bp_r^q+r_1,T / rr \ nr \ llm\ap,q) — ^p-r.g+r

3.2. Hochschild-Serre spectral sequence. Let g be a perfect Lie algebra and let u beits universal central extension in the category (Lie) with kernel \] = H2(fl). Since \] is anabelian Lie algebra on which g acts trivially, the i?2-terms of the Hochschild-Serre spectralsequence are given by

q - Hp(fl, H,(f,)) = A«(fj) % Hp(fl)

10

Theorem 3.1. Let u be the universal central extension of a perfect Lie algebra g. Then onehas an exact sequence

from whence the (non-natural) isomorphism

H3(u)^S2(H2( f l)) © H3(fl)

where S denotes the symmetric functor.

Proof. We have to compute separately the four terms E™q with p+ q = 3. Firstly recallthat the Hochschild-Serre spectral sequence is derived from the filtration (F^)p>0 where Fv

a

is the subspace of An(u) generated by the elements x\ A- • • l\xn with at least (n — p) factorsin \).

i) The term EQ°3. One has EQ 3 = A3(()). Let us determine the image lm(d\ 2) =BQ 3/-BQ 3 . Since u is perfect, one has

£0,3 = ^3 n d{Fl) = A3(o) n d{Fl) = A3(o).

On the other hand, we have

Therefore \m(d\ 2) = A3(()); from whence we deduce

poo ~ . . . ~ p3 — r.

ii) The term E^°2- Since the Lie algebra g is perfect, it is clear that

E™2 ^ • • • ^ E2l2 = 0.

iii) T h e t e r m E^. One has E\x ^ f) <8> H2(fl) = f)®2. Firstly let us determine the

image Imf^2, x) = B\ 2/^h 2- Since u is perfect, we have

£0,2 = 2̂° n d{Fl) = A2([)) n d{Fl) = A2([)).

On the other hand, we have

Blfi = F2° n d(Fl) = A2([)) n {0} = 0.

Therefore lm{d221) = A2([)); from whence ke r (^ 1 ) ^ S2([)).

Now we have to calculate \m(d\ 0) = B2 l/'{Z\ 2 + B\ t) with

zl)2 = Fi n d-\Fl) = Fi n d~l(A2([))),^2,i = ^32nrf(F3).

As vector space we haveF4

4=A4(u) = F | + A4(fl)

11

which yieldsd{Ft) = d{Fl) + rf(A4(fl)).

Thus it is clear, that

-^2,1 = (-^2,1 + ^1,2)-

From whence we deduce that lm(d\ 0) = 0. And then we have

E 2 l = ••• = E 2 l = h [\]).

iv) The term E^°o. Since E\ x = 0 (perfectness of g), we have

v) Conclusion. Since K is a field, the exact sequence (hence the isomorphism) ofProposition 3.1 is clear taking into account the above computations. •

3.3. Leibniz spectral sequence. Recall that for any Leibniz algebra Q, any two-sidedideal H in Q and any semi-representation M of Q, the filtration (-F^)p>o defined by

pv .—® if n <p,

gives rise to a spectral sequence (Er(H,M))r>i which converges to the Leibniz homologyHL*(£,M) (see [4]). Moreover, if the adjoint diagonal action of Q/fL on RLq(fL1M) istrivial, then one has the isomorphisms

E l ^ M) £* HLg(7^, M) ® HLp_!(g, G/U), p > 2.

Here we let U{= Q) be the universal central extension of a perfect Leibniz algebra CDenote by % := HL2(£) the kernel of this universal central extension. Since the Leibnizalgebra L = lAjH acts trivially on 7i, it also acts trivially on HLg(%) = T~L®q for anyinteger q > 0. Therefore, by the perfectness of £ (i.e., HLi(£) = 0), the spectral sequence(Er(7i, K))r>i is characterized by

£02ig(^,K) ^U®\ Elq(H,K) =0,

Elq(n,K)*n®q®nLp-1(u,c), P>2.

Theorem 3.2. Let hi be the universal central extension of a perfect Leibniz algebra C Thenone has an exact sequence

0 -^ HL2 (£) ®2 -^ HL3 (U) -^ HL3 (£) -^ 0,

/ram whence the isomorphism

HL 3 (^ )^HL 2 (£ )® 2 © HL3(£).

Proof. In order to obtain HL3(ZY), we are led to compute HLi(ZY,£) and HL2(^/,£). Tothis end, we will consider the spectral sequence (Er(%,C))ryi.

12

Lemma 3.3. One has an isomorphism

HLi(W,£) = HL2(£)

and an exact sequence

0 -^ HL2 (£) ®2 -^ HL3 (W) -^ HL3 (£) -^ 0.

i) From the perfectness of U, one easily checks that

And sincezl3 = n®3nd-1(o) = n®3,

we have £^3(?^,K) = 0.ii) We already know that E\ 2{H, K) = 0 (perfectness of C).iii) From the perfectness of 14, we also get

And since

we deduce that

But we know that

from whence we have E2 x(H, K) = 0.iv) Since E\ i{T~L, K) = 0 (perfectness of £.), one easily checks that

E™0(U,K)^---^ E23fi(H,K)£* HL 2 {U,L) .

v) Conclusion. Therefore there exists an exact sequence

0 -^ HL2 (£) ®2 -> HL3 (U) -^ HL3 (£)-)• 0,

from whence the isomorphism

HL2(£)®2 © HL3(£). D

Proof of the Lemma. Recall that we are using the spectral sequence (Er(7i,C))r>i.

i) Computation of HLi(ZY,£). One easily checks that

^ £ ® 7^/[£ ® 7̂ , £] ^ (£/[£, £]) ® U = 0,

E2fi{U,C) ^HL!(£,HL0(^

13

Therefore we have

HLi(W,£) ^ H L 2 ( £ ) ^ f t .

ii) Computation of HL2(ZY,£). As before we have

F2j 2 (ft, C) =* HL0 (£, HL2 (ft, £)) =* HL0 (£, £ ® HL2 (ft))

® 2 ® 2 ] ) » ft®2 = o,HLi (ft))

Here one also needs to return to the characterization of

We have

Therefore we obtain

And it is clear that

In order to determine Ff 0, we use the following commutative diagram

d'2 d\ 0 d'J,

I I I

Denoting by a the image in C = li/ft of an element a £li, one easily checks that

Therefore, the long exact sequence in homology yields the exact sequence

® 2 ® 2 , £ ] ) ® ft A F2j 0(ft ,£) - ^ HL3(£) -^ 0.

14

On the other hand, ones knows that

But here we have

z\A = F\ n r 1 ^ ) = (c®2®u)nd-1(c®?{).

Thus we have Im(i) C Z\ 1'1 from whence we deduce that

E^0(n,£) =* • • • ^ Elo(H,C) ^HL3(£).

Therefore the Lemma is clear. •

3.4. Interpretation of the natural maps. From now on, let g be a perfect Lie algebraand let il (resp. u) be its universal central extension in the category (Leib) (resp. (Lie)).Since the Leibniz algebra il is also the universal central extension of u, the Proposition 3.2yields the isomorphisms

HL3(il) = HL2(fl)®2 © HL3(fl) = HL2(u)®2 © HL3(u),

and we know that

HL2(u)^ker(HL2(fl) -» H2(fl)) = ker(il -» u =* iiLie).

Therefore we can state the characterization of the natural maps HL3(il) —> HL3(u) andHL3(u)^H3(u).

Theorem 3.4. Let il (resp. u) be the universal central extension in the category (Leib)(resp. (Lie)) of a perfect Lie algebra g. Then we have a (non-natural) commutative diagram

HL3(il) —^

IHL3(u) —^-

IH3(u) — ^ •

-+ HL2(fl)®2

I-^ HL2(fl)®

2/K®2

I-^ S2(H2(fl))

©

©

©

HL3(fl)

1HL3(fl)

IH3(fl)

whereK :=ker(HL2(g) >̂ H2(fl)) = ker(il >̂ u ^ ilLje) ^HL2(u).

Proof. Recall that the natural map HL*(il) —> HL*(u) (resp. HL*(u) —> H*(u)) isinduced by the canonical projection

i l ® * ^ ( i l L j e ) ® * (resp. U ® * ^ A » ) .

Given a Lie algebra [, one has a commutative diagram of ((little» complexes

> ?? > S2([) > 0

i I~^ ~^ ~^

A3([) > A2([)

15

which yields in homology the exact sequence

(t) HL3([) -^ H3([) -^ S2 ( [ ) / - -^ HL2 ([) -^ H2 ([) -^ 0.

Here "~" stands for the relations generated in S ([) by the elements

([%, y], z ) - ([x, z ] , y ) - ( x , [y, z ] ) , V x , y , z G L

Applied to the Lie algebras u and g, we obtain the diagram

HL3(u) > H3(u) > S2(u)/ > K > 0 > 0

N^ N^ N^ N^

HL3(fl) > H3(fl) • S2(fl)/ > HL2(fl) • H2(fl) > 0.

Therefore it is clear that HL3 (g) n /i' ®2 = 0; from whence the computation of HL3(u). Thecommutative diagram is now obvious taking into account the previous calculations and thelast diagram. •

3.5. Remark. T. Pirashvili studies in general the kernel of the canonical map

This gives rise to a notion of relative homology which enables him to construct a long exactsequence generalizing the sequence (f) (see [13]).

4. A finer Leibniz spectral sequence

There exists another filtration {FL%)V>Q of the Leibniz complex, more efficient than theabove one. The vector space FLp

n is the submodule of M (£)Q®n generated by the elementsm® x\ ® • • • ® xn such that at least (n — p) factors X{ are in %. Obviously the filtration{F*)v>o is a subfiltration of (FL*)p>0. Nevertheless the EL2-tervas of the spectral sequencederived from this finer filtration are slightly more complicated to determine. It is expectedthat if M = K and if the adjoint diagonal action of QfH on HL*(%) is trivial, then theEL -terms are given by

Here • stands for the free product of graded modules that is, the non-commutative analogueof the classical tensor product <8>, which arises in the isomorphism

T(V © V) ^T(V)*T(V').

To be more precise, EL^q is the direct sum of the components

Xil 0 Yi2 0 Xis <g> Yj4 <g> • • • , ii + i2 + «3 + h H = p + q

such that eitherX* = EL^g/Ti) and Ym = H

orand Ym = YiLm{

16

In particular, this will give an idle way of getting the Kiinneth-style formula for theLeibniz homology (see [9], [12]).

Acknowledgements

I would like to express my gratitude to Professors J.-L. Loday and A. O. Kuku for veryenlightening discussions and helpful suggestions. Also, I warmly thank all those who hadclosely contributed for my presence in this marvellous and stimulating research environmentwhich is the Abdus Salam International Centre for Theoretical Physics of Trieste (Italy). Aspecial glance at Miss Sharon Laurenti for her «raison d'etre».

REFERENCES

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[2] Chevalley C. and Eilenberg S., Cohomology theory of Lie groups and Lie algebras, Trans. A.M.S. 63(1948), 85-124.

[3] Garland H., The arithmetic theory of loop groups, Publ. I.H.E.S. 52 (1980), 5-136.[4] Gnedbaye A. V., Suite spectrale d'une extension d'algebres de Leibniz, C. R. Acad. Sci. Paris, Serie I

324 (1997), 1327-1332.[5] Hochschild G. and Serre J.-P., Cohomologie of Lie algebras, Annal. Math. 57 (1953), 591-603.[6] Kassel Ch. and Loday J.-L., Extensions centrales des algebres de Lie, Annal. Inst. Fourier 32 (1982),

119-142.[7] Loday J.-L., Cyclic homology, vol. 301, Grand, math. Wiss., Springer-Verlag, 1992.[8] Loday J.-L., Une version non commutative des algebres de Lie: les algebres de Leibniz, L'Enseignement

Math. 39, (1993), 269-293.[9] Loday J.-L., Runneth-style formula for the homology of Leibniz algebras, Math. Zeit. 221 (1996), 41-47.[10] Loday J.-L. and Pirashvili T., Universal enveloping algebras of Leibniz algebras and (co)homology,

Math. Annal. 296 (1993), 139-158.[II] Me Cleary J., User's guide to spectral sequences, vol. 12, Mathematics Lecture Series, Publish or Perish,

Inc., 1985.[12] Oudom J.-M., La diagonale en homologie des algebres de Leibniz, C. R. Acad. Sci. Paris, Serie I 320

(1995), 1165-1170.[13] Pirashvili T., On Leibniz homology, Ann. Inst. Fourier 44 (1994), 401-411.