NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS - Mathematics | Illinois

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NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS
1 2 3
N. INASSARIDZE , E. KHMALADZE AND M. LADRA
1;2 A.Razmadze Mathematical Institute, Georgian Academy of Sciences, M.Alexidze St. 1,
Tbilisi 380093, Georgia, e-mail: finas, [email protected]
3 Departamento de Algebra, Facultad de Matema(cid:19)ticas, Universidad de Santiago de Compostela,
Santiago de Compostela 15782, Spain, e-mail: [email protected]
Abstract: Non-abelianhomologyofLiealgebraswithcoe(cid:14)cientsinLiealgebrasisconstructedandstudied,
generalising the classical Chevalley-Eilenberg homology of Lie algebras. The relation of cyclic homology
with Milnor cyclic homology of non-commutative associative algebras is established in terms of the long
exactnon-abelianhomologysequenceof Liealgebras. Some explicit formulasforthe secondand the third
non-abelian homology of Lie algebras are obtained. Using the generalised notion of the Lie algebra of
derivations, we introduce the second non-abelian cohomology of Lie algebras with coe(cid:14)cients in crossed
modules and extend the seven-term exact non-abelian cohomology sequence of Guin to nine-term exact
sequence.
AMS 2000 Mathematics subject classi(cid:12)cation: 17B40,17B56, 18G10, 18G50.
Keywords: Non-abelian tensor product, non abelian (co)homology of Lie algebras, cyclic homology.
0. Introduction
The non-abelian homology of groups with coe(cid:14)cients in groups was constructed and inves-
tigatedin[21,20],usingthenon-abeliantensorproductofgroupsofBrownandLoday[5,6]
and its non-abelian left derived functors. It generalises the classical Eilenberg-MacLane
homology of groups and extends Guin’s low dimensional non-abelian homology of groups
with coe(cid:14)cients in crossed modules [13], having an interesting application to the algebraic
K-theory of non-commutative local rings [13, 21].
In [17, 18, 19] H. Inassaridze developed a non-abelian cohomology theory previously
de(cid:12)ned by Guin in low dimensions [13] which di(cid:11)ers from the classical (cid:12)rst non-abelian
cohomology pointed set of Serre [24] and from the setting of various papers on non-abelian
1
2 N.INASSARIDZE,E. KHMALADZEAND M. LADRA
cohomology [10, 4, 9] extending the classical exact non-abelian cohomology sequence from
lower dimensions [24] to higher dimensions.
The purpose of this paper is to set up a similar non-abelian (co)homology theory for
Lie algebras and mainly dedicated to state and prove several desirable properties of this
(co)homology theory.
In [12] Ellis introduced and studied the non-abelian tensor product of Lie algebras which
isaLiestructuralandpurelyalgebraicanalogueofthenon-abeliantensorproductofgroups
of Brown and Loday [5, 6], arising in applications to homotopy theory of a generalised Van
Kampen theorem.
Applying this tensor product of Lie algebras, Guin de(cid:12)ned the low-dimensional non-
abelian homology of Lie algebras with coe(cid:14)cients in crossed modules [14].
We construct a non-abelian homology H (M;N) of a Lie algebra M with coe(cid:14)cients
(cid:3)
in a Lie algebra N as the non-abelian left derived functors of the tensor product of Lie
algebras, generalising the classical homology of Lie algebras and extending Guin’s non-
abelian homology of Lie algebras [14]. We give an application of our long exact homology
sequence to cyclic homology of associative algebras, correcting the result of [14]. In fact,
for a unital associative (non-commutative) algebra A we obtain a long exact non-abelian
homology sequence
(cid:1)(cid:1)(cid:1) (cid:0)! H (A;V(A);[A;A]) (cid:0)! H (A;V(A)) (cid:0)! H (A;[A;A])
2 2 2
(cid:0)! H (A;V(A);[A;A]) (cid:0)! H (A;V(A)) (cid:0)! H (A;[A;A])
1 1 1
(cid:0)! HC (A) (cid:0)! HCM(A) (cid:0)! [A;A]=[A;[A;A]] (cid:0)! 0:
1 1
Following[14]andusingideasfrom[17],weintroducethesecondnon-abeliancohomology
H2(R;M) of a Lie algebra R with coe(cid:14)cients in a crossed R-module (M;(cid:22)), generalising
theclassical secondcohomologyofLiealgebras. Then, foracoe(cid:14)cientshortexactsequence
of crossed R-modules
0 (cid:0)! (L;0) (cid:0)! (M;(cid:22)) (cid:0)! (N;(cid:23)) (cid:0)! 0 ;
having a module section over the ground ring, we give a nine-term exact non-abelian
cohomology sequence
0 (cid:0)! H0(R;L) (cid:0)! H0(R;M) (cid:0)! H0(R;N) (cid:0)! H1(R;L) (cid:0)! H1(R;M)
(cid:0)! H1(R;N) (cid:0)! H2(R;L) (cid:0)! H2(R;M) (cid:0)! H2(R;N) ;
extending the seven-term exact cohomology sequence of Guin [14] which exists under the
aforementioned additional necessary condition on the coe(cid:14)cient sequence of crossed mod-
ules.
Further generalisations of non-abelian cohomology of Lie algebras is possible pursuing
the line of [18, 19], in particular to the direction of making a de(cid:12)nition in any dimension
and for a wider class of coe(cid:14)cients.
In future papers we hope to return in detail analysis of the non-abelian homology of
Lie algebras of matrices in connection with cyclic homology and de Rham cohomology
following ideas of Loday and Quillen [23].
NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS 3
Notations and Conventions. We denote by (cid:3) a unital commutative ring unless other-
wise stated. We shall use the term Lie algebra to mean a Lie algebra over (cid:3) and [ ; ] and
j j to denote the Lie bracket and the coset of the quotient Lie algebra respectively. We
denote the category of Lie algebras over (cid:3) by Lie.
We mean under the classical (co)homology of a Lie algebra M with coe(cid:14)cients in an M-
module N, the (co)homologygroups in thesense ofChevalley-Eilenberg (see e.g. [7])which
are the homology of the complexes obtained by applying the functors Hom ((cid:0);N) and
U(M)
(cid:0)(cid:10) N to the following standard complex of U(M)-modules
U(M)
@ @ @ @ (cid:15)
(cid:1)(cid:1)(cid:1) (cid:0)! V (M) (cid:0)! (cid:1)(cid:1)(cid:1) (cid:0)! V (M) (cid:0)! V (M) (cid:0)! (cid:3) ;
n 1 0
where U(M) is the universal enveloping algebra of M, V (M) = U(M)(cid:10) E (M), n (cid:21) 0,
n (cid:3) n
E (M) = M^ (cid:1)(cid:1)(cid:1)^ M , n (cid:21) 1 and E (M) = (cid:3) and the chain boundary is given by the
n (cid:3) (cid:3) 0
| n(cid:0){tizmes }
formula
n
@hx ;:::;x i = ((cid:0)1)i+1x hx ;:::;x^;:::;x i
1 n P i 1 i n
i=1
+ ((cid:0)1)i+jh[x ;x ];x ;:::;x^;:::;x^;:::;x i ;
P i j 1 i j n
1(cid:20)i<j(cid:20)n
and (cid:15)h i = 1.
1. The non-abelian tensor product
Let P and M be two Lie algebras. By an action of P on M we mean a (cid:3)-bilinear map
P (cid:2)M ! M, (p;m) 7! pm satisfying the following conditions:
[p;p0]m = p(p0m)(cid:0)p0(pm); p[m;m0] = [pm;m0]+[m;pm0]
for all m; m0 2 M and p; p0 2 P. For example, if P is a subalgebra of some Lie algebra
Q, and if M is an ideal in Q, then Lie multiplication in Q yields an action of P on M.
Now we give the de(cid:12)nition of the tensor product of Lie algebras due to Ellis [12] (see
also [14], [8]). Let M and N be two Lie algebras acting on each other. The tensor product
M (cid:10)N of the Lie algebras M and N is the Lie algebra generated by the symbols m(cid:10)n,
m 2 M, n 2 N, and subject to the following relations:
(i) (cid:21)(m(cid:10)n) = (cid:21)m(cid:10)n = m(cid:10)(cid:21)n,
(ii) (m+m0)(cid:10)n = m(cid:10)n+m0 (cid:10)n,
m(cid:10)(n+n0) = m(cid:10)n+m(cid:10)n0,
(iii) [m;m0](cid:10)n = m(cid:10)(m0n)(cid:0)m0 (cid:10)(mn),
m(cid:10)[n;n0] = (n0m)(cid:10)n(cid:0)(nm)(cid:10)n0,
(iv) [(m(cid:10)n);(m0 (cid:10)n0)] = (cid:0)(nm)(cid:10)(m0n0)
for all (cid:21) 2 (cid:3), m;m0 2 M, n;n0 2 N.
4 N.INASSARIDZE,E. KHMALADZEAND M. LADRA
Suppose (cid:30) : M ! A, : N ! B are Lie homomorphisms, A, B act on each other, and
(cid:30), preserve the actions in the following sense:
(cid:30)(nm) = (n)(cid:30)(m) ; (mn) = (cid:30)(m) (n) ; m 2 M ; n 2 N :
Then by [12] there is a unique homomorphism (cid:30) (cid:10) : M (cid:10) N ! A (cid:10) B such that
((cid:30)(cid:10) )(m(cid:10)n) = (cid:30)(m)(cid:10) (n) for all m 2 M, n 2 N. Furthermore, if (cid:30), are onto, so
also is (cid:30)(cid:10) .
ThetensorproductofLiealgebrasissymmetricinthesenseoftheisomorphismM(cid:10)N !
N (cid:10)M given by m(cid:10)n 7! (cid:0)n(cid:10)m [12].
A precrossed P-module (M;(cid:22)) is a Lie homomorphism (cid:22) : M ! P together with an
action of P on M satisfying the following condition:
(cid:22)(pm) = [p;(cid:22)(m)] :
If in addition the precrossed module (M;(cid:22)) satis(cid:12)es the Pei(cid:11)er identity:
(cid:22)(m)m0 = [m;m0] ;
then it is said to be a crossed P-module. Note that for a crossed module (M;(cid:22)) the image
of (cid:22) is necessarily an ideal in P and the kernel of (cid:22) is a P-invariant ideal in the center
of M. Moreover the action of P on Ker(cid:22) induces an action of P=Im(cid:22) on Ker(cid:22), making
Ker(cid:22) a P=Im(cid:22)-module.
In [12] the results on the tensor product M (cid:10) N are obtained assuming the actions of
M and N on each other compatible, i.e.
(1) (nm)n0 = [n0;mn] and (mn)m0 = [m0;nm]
for all m;m0 2 M and n;n0 2 N. This is the case, for example, if (M;(cid:22)) and (N;(cid:23)) are
crossed P-modules, M and N act on each other via the action of P. These compatibility
conditions are not assumed to hold except the following
Proposition 1. Let M and N be Lie algebras acting on each other such that the compat-
ibility conditions (1) hold. Then there is a natural isomorphism of (cid:3)-modules
(cid:24)
M (cid:10)N = (M (cid:10) N)=D(M;N) ;
(cid:3)
where D(M;N) is the (cid:3)-submodule of M (cid:10) N generated by the elements
(cid:3)
[m;m0](cid:10)n(cid:0)m(cid:10)(m0n)+m0 (cid:10)(mn) ;
m(cid:10)[n;n0](cid:0)(n0m)(cid:10)n+(nm)(cid:10)n0 ;
(nm)(cid:10)(mn) ;
(nm)(cid:10)(m0n0)+(n0m0)(cid:10)(mn) ;
[nm;n0m0](cid:10)(m00n00)+[n0m0;n00m00](cid:10)(mn)+[n00m00;nm](cid:10)(m0n0)
for all m; m0; m00 2 M and n; n0; n00 2 N.
NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS 5
Proof. Let us introduce in the (cid:3)-module (M (cid:10) N)=D(M;N) a Lie structure by the fol-
(cid:3)
lowing formula
[m(cid:10)n;m0 (cid:10)n0] = (cid:0)(nm)(cid:10)(m0n0) :
To show that this multiplication could be extended from generators to any elements of
(M (cid:10) N)=D(M;N) we have to check that it is compatible with the de(cid:12)ning relations of
(cid:3)
(M (cid:10) N)=D(M;N). For instance,
(cid:3)
[[m;m0](cid:10)n;m00 (cid:10)n00] = (cid:0)(n[m;m0])(cid:10)(m00n00) = (cid:0)[nm;m0](cid:10)(m00n00)(cid:0)[m;nm0](cid:10)(m00n00)
= (cid:0)((m0n)m)(cid:10)(m00n00)+((mn)m0)(cid:10)(m00n00) = [m(cid:10)(m0n);m00(cid:10)n00](cid:0)[m0(cid:10)(mn);m00(cid:10)n00] ;
[m00 (cid:10)n00;m(cid:10)[n;n0]] = (cid:0)(n00m00)(cid:10)(m[n;n0]) = (cid:0)(n00m00)(cid:10)[mn;n0](cid:0)(n00m00)(cid:10)[n;mn0]
= (n00m00)(cid:10)((nm)n0)(cid:0)(n00m00)(cid:10)((n0m)n) = [m00 (cid:10)n00;(n0m)(cid:10)n](cid:0)[m00 (cid:10)n00;(nm)(cid:10)n0] ;
[(nm)(cid:10)(mn);m0 (cid:10)n0] = (cid:0)((mn)(nm))(cid:10)(m0n0) = (cid:0)[nm;nm](cid:10)(m0n0) = 0 ;
[m00 (cid:10)n00;(nm)(cid:10)(m0n0)] = (cid:0)(n00m00)(cid:10)((nm)(m0n0)) = (cid:0)(n00m00)(cid:10)[m0n0;mn]
= (n00m00)(cid:10)[mn;m0n0] = (n00m00)(cid:10)((n0m0)(mn)) = (cid:0)[m00 (cid:10)n00;(n0m0)(cid:10)(mn)] :
(cid:3)
Now we investigate for the tensor product of Lie algebras the properties of compatibility
with the direct limits of Lie algebras and the right exactness which will be used in the
sequel.
Proposition 2. Let fM ; (cid:30)(cid:12); (cid:11) (cid:20) (cid:12)g be a direct system of Lie algebras. Let N be a Lie
(cid:11) (cid:11)
algebra and let for every (cid:11) the Lie algebras M , N act on each other and the homomor-
(cid:11)
phisms (cid:30)(cid:12) preserve the actions. Then there is a natural isomorphism of Lie algebras
(cid:11)
(cid:24)
(limfM g)(cid:10)N = limfM (cid:10)Ng :
(cid:11) (cid:11)
(cid:0)! (cid:0)!
(cid:11) (cid:11)
Proof. Let us de(cid:12)ne the actions of limfM g and N on each other by the following way:
(cid:11)
(cid:0)!
(cid:11)
jm(cid:11)jn = m(cid:11)n and njm j = jnm j
(cid:11) (cid:11)
for all m 2 M , n 2 N. It is easy to check that the actions are correctly de(cid:12)ned.
(cid:11) (cid:11)
Now de(cid:12)ne a Lie homomorphism
f : (limfM g)(cid:10)N (cid:0)! limfM (cid:10)Ng
(cid:11) (cid:11)
(cid:0)! (cid:0)!
(cid:11) (cid:11)
by f(jm j(cid:10)n) = jm (cid:10)nj. If jm j = jm j, then there exists (cid:13) (cid:21) (cid:11);(cid:12) such that (cid:30)(cid:13)(m ) =
(cid:11) (cid:11) (cid:11) (cid:12) (cid:11) (cid:11)
(cid:30)(cid:13)(m ). Thus ((cid:30)(cid:13) (cid:10)1 )(m (cid:10)n) = ((cid:30)(cid:13) (cid:10)1 )(m (cid:10)n). Hence jm (cid:10)nj = jm (cid:10)nj and f
(cid:12) (cid:12) (cid:11) N (cid:11) (cid:12) N (cid:12) (cid:11) (cid:12)
is correctly de(cid:12)ned. Commutativity of f with the de(cid:12)ning relations of the tensor product
of Lie algebras is easy and will be omitted.
6 N.INASSARIDZE,E. KHMALADZEAND M. LADRA
On the other hand the canonical Lie homomorphisms
(cid:30) : M (cid:10)N (cid:0)! (limfM g)(cid:10)N
(cid:11) (cid:11) (cid:11)
(cid:0)!
(cid:11)
given by (cid:30) (m (cid:10)n) = jm j(cid:10)n induce a Lie homomorphism
(cid:11) (cid:11) (cid:11)
g : limfM (cid:10)Ng (cid:0)! (limfM g)(cid:10)N
(cid:11) (cid:11)
(cid:0)! (cid:0)!
(cid:11) (cid:11)
and one can easily check that fg, gf are identity maps. (cid:3)
(cid:30)
Proposition 3. Suppose 0 ! M0 ! M ! M00 ! 0 is a short exact sequence of Lie
algebras, N is an arbitrary Lie algebra acting on M0, M and M00; the Lie algebras M0,
M, M00 act on N and (cid:30), preserve these actions. Then there is an exact sequence of Lie
algebras
M0 (cid:10)N (cid:0)! M (cid:10)N (cid:0)! M00 (cid:10)N (cid:0)! 0
Proof. Similarly to the proof of Proposition 9 [12], since it does not use compatibility
(cid:3)
conditions (1).
2. Construction of non-abelian homology
We begin this section by recalling the well-known construction of the free Lie algebra on
some (cid:3)-module.
Let M be a (cid:3)-module. Let A (M) = M, A (M) = A (M) (cid:10) A (M) and
1 k P i (cid:3) k(cid:0)i
0<i<k
A(M) = A (M). The inclusion maps A (M) (cid:10) A (M) ! A (M) give rise to a
P k i (cid:3) k i+k
0<k
non-associative multiplication on A(M) turning it into an algebra over (cid:3).
Let B(M) be the two-sided ideal of A(M) generated by the elements
xx and x(yz)+y(zx)+z(xy) ;
for all x; y; z 2 A(M).
We obtain the Lie algebra F(M) = A(M)(cid:30)B(M) which is the free Lie algebra on the (cid:3)-
module M satisfying the following universal property: there is a natural (cid:3)-homomorphism
i : M ! F(M) such that for any Lie algebra L and a (cid:3)-homomorphism (cid:11) : M ! L there
exists a unique Lie homomorphism (cid:20) : F(M) ! L such that (cid:20)i = (cid:11).
Let N be a Lie algebra and (cid:11) : M ! Der(N) a (cid:3)-homomorphism, where Der(N)
is the Lie algebra of derivations of N. Then there exists a unique Lie homomorphism
(cid:20) : F(M) ! Der(N) such that (cid:20)i = (cid:11), that means there is an action of the Lie algebra
F(M) on the Lie algebra N.
Now if in addition M is an N-module, then the module action of N on M yields an N-
module structure on A (M): if x(cid:10)y 2 A (M)(cid:10) A (M) and n 2 N then, inductively,
k i (cid:3) k(cid:0)i
we de(cid:12)ne
n(x(cid:10)y) = nx(cid:10)y+x(cid:10)ny ;
NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS 7
and this extends linearly to an action of n on an arbitrary element of A (M). The action
k
of N on A (M) extends linearly to an action of N on A(M), making A(M) into an N-
k
module. Since B(M) is N-invariant, the action of N on A(M) induces a Lie action of N
on F(M).
Let A denote, for a (cid:12)xed Lie algebra N, the category whose objects are all Lie algebras
N
M together with an action of M on N by derivations of N and an action of N on M by
derivations of M. Morphisms in the category A are all Lie homomorphisms (cid:11) : M ! M0
N
preserving the actions, namely (cid:11)(nm) = n(cid:11)(m) and mn = (cid:11)(m)n for all m 2 M, n 2 N.
Let F : A ! A be the endofunctor de(cid:12)ned as follows: for an object M of A , let
N N N
F(M)denotethefreeLiealgebraontheunderlying(cid:3)-moduleM withtheabove-mentioned
actions of N on F(M) and F(M) on N; for a morphism (cid:11) : M ! M0 of A , let F((cid:11)) be
N
the canonical Lie homomorphism from F(M) to F(M0) induced by (cid:11).
Let (cid:28) : F ! 1A be the obvious natural transformation and let (cid:14) : F ! F2 be
N
the natural transformation induced for every M 2 obA by the natural inclusion of (cid:3)-
N
modules M ! F(M). We obtain the cotriple F = (F; (cid:28); (cid:14)). Let P be the projective class
in the category A induced by the cotriple F. It is easy to see that in the category A
N N
there exist (cid:12)nite limits. Therefore every object M of the category A has a P-projective
N
pseudosimplicial resolution (F ;d0;M) (see [26, 16]).
(cid:3) 0
Let T : A ! Lie be a covariant functor. Applying T dimension-wise to F yields the
N (cid:3)
simplicial Lie algebra TF . De(cid:12)ne the k-th derived functor LPT : A ! Lie, k (cid:21) 0, of
(cid:3) k N
the functor T relative to the projective class P as the k-th homotopy of T F . Note that
(cid:3)
P
L T (M), k (cid:21) 1, is an abelian Lie algebra and will be thought as a (cid:3)-module. Hence,
k
forgetting the Lie algebra structure, the result in [26] implies that there is an isomorphism
of functors
P (cid:24) F
L T = L T ; k (cid:21) 0 ;
k k
F
where L T is k-th cotriple derived functor of the functor T de(cid:12)ned using the cotriple
k
resolution (F(cid:3)(M);(cid:28)(M);M) of an object M in the category A
N
F(cid:3)(M) (cid:17) (cid:1)(cid:1)(cid:1)!... Fk+1(M) !d...k0 (cid:1)(cid:1)(cid:1)!!d20 F2(M) !!d10 F1(M) ;
! ! ! d1
dkk d22 1
Fk+1(M) = F(Fk(M)), dk = Fi(cid:28)Fk(cid:0)i(M), sk = Fi(cid:14)Fk(cid:0)i(M).
i i
The next lemma is useful. The proof is easy and left to the reader.
Lemma 4. A morphism (cid:11) : M ! M0 of the category A is a P-epimorphism if and only
N
if there exists a (cid:3)-linear splitting (cid:12) : M0 ! M which preserves the actions of N, i.e.
(cid:12)(nm) = n(cid:12)(m) for all m 2 M, n 2 N.
The non-abelian tensor product of Lie algebras de(cid:12)nes a covariant functor (cid:0)(cid:10)N from
the category A to the category Lie. Consider the left derived functors LP((cid:0)(cid:10)N), k (cid:21) 0,
N k
of the functor (cid:0)(cid:10)N relative to the projective class P.
8 N.INASSARIDZE,E. KHMALADZEAND M. LADRA
Proposition 5. Let M be a Lie algebra and N a module over the Lie algebra M, then
there are natural isomorphisms
P (cid:24)
L ((cid:0)(cid:10)N)(M) = H (M;N) ; k (cid:21) 1 ;
k k+1
(cid:24) (cid:24)
Ker(cid:23) = H (M;N) ; Coker(cid:23) = H (M;N) ;
1 0
where N is thought as an abelian Lie algebra acting trivially on M, (cid:23) : M (cid:10)N ! N is a
Lie homomorphism given by (cid:23)(m(cid:10)n) = mn, m 2 M, n 2 N.
Proof. Let Lie denote the category of Lie algebras over M and Di(cid:11) : Lie ! U(M)(cid:0)
M M M
mod (category of U(M)-modules) a functor given by
Di(cid:11) (W) = I(W)(cid:10) U(M) ;
M U(W)
where U(M) and U(W) are the universal enveloping algebras ofM and W respectively and
F
I(W)istheaugmentationideal. ByProposition13[8]L ((cid:0)(cid:10)N)(M) areisomorphic tothe
(cid:3)
values of the non-abelian left derived functors of the functor Di(cid:11) ((cid:0))(cid:10) N : Lie !
M U(M) M
(cid:3) (cid:0) mod (category of (cid:3)-modules) for the object 1 of the category Lie which give
M M
the classical homology H (M;N) of Lie algebras with the usual dimension shift, similarly
(cid:3)
to the cases of group (co)homology and Hochschild (co)homology described as cotriple
(cid:3)
(co)homology [3, 2].
Using this proposition we make the following
De(cid:12)nition 6. Let M and N be Lie algebras acting on each other. De(cid:12)ne the non-abelian
homology of M with coe(cid:14)cients in N by setting
P
H (M;N) = L ((cid:0)(cid:10)N)(M) ; k (cid:21) 2 ;
k k(cid:0)1
H (M;N) = Ker(cid:23) ; H (M;N) = Coker(cid:23) ;
1 0
where (cid:23) : M (cid:10) N ! N(cid:30)H, (cid:23)(m (cid:10) n) = jmnj, and H is the ideal of the Lie algebra N
generated by the elements (nm)n0 (cid:0)[n0;mn] for all m 2 M, n;n0 2 N.
Remark. (a) It is clear that H (M;N), k (cid:21) 2, are only (cid:3)-modules, when H (M;N) and
k 1
H (M;N) are Lie algebras. If the actions of M and N satisfy the compatibility conditions
0
(1), then H (M;N) is also an abelian Lie algebra.
1
(b)Let N be a crossed M-module, then H (M;N) and H (M;N) coincides with zero and
0 1
(cid:12)rst non-abelian homology (cid:3)-modules of the Lie algebra M with coe(cid:14)cients in the crossed
M-module N introduced by Guin [14].
One could de(cid:12)ne another non-abelian homology theory of Lie algebras using the non-
abelian left derived functors of the non-abelian tensor product relative to the cotriple over
sets which coincides with our theory for Lie algebras being free (cid:3)-modules.
3. Some properties of non-abelian homology
In this section we investigate the non-abelian homology of Lie algebras in various aspects,
establishing its some functorial properties.
NON-ABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS 9
Now several long exact non-abelian homology sequences with respect to both variables
will be given.
Theorem 7. Let (cid:11) : N ! N0 be a surjective Lie homomorphism, M an arbitrary Lie
algebra acting on N and N0 which act on M and (cid:11) preserve the actions. Then there is a
long exact sequence of non-abelian homology
(cid:1)(cid:1)(cid:1) (cid:0)! H (M;N0) (cid:0)(cid:14)!3 H (M;N;N0) (cid:0)j!2 H (M;N) (cid:0)i!2 H (M;N0)
3 2 2 2
(2) (cid:0)(cid:14)!2 H (M;N;N0) (cid:0)j!1 H (M;N) (cid:0)i!1 H (M;N0) (cid:0)(cid:14)!1 H (M;N;N0)
1 1 1 0
(cid:0)j!0 H (M;N) (cid:0)i!0 H (M;N0) (cid:0)! 0 ;
0 0
where
Hk(M;N;N0) = (cid:25)k(cid:0)1(Ker(1F(cid:3)(M) (cid:10)(cid:11))) ; k (cid:21) 2 ;
fKer(1 (cid:10)(cid:11))\(d0 (cid:10)1 )(cid:0)1(Ker(1 (cid:10)(cid:11))\Ker(cid:23))g
H (M;N;N0) = F1(M) 0 N M ;
1 (d1 (cid:10)1 )(Ker(1 (cid:10)(cid:11))\Ker(d1 (cid:10)1 ))
1 N F2(M) 0 N
H (M;N;N0) = Ker(cid:11)=(cid:23)(Ker(1 (cid:10)(cid:11))) ;
0 M
e
(F(cid:3)(M);d0;M) is the F cotriple resolution of the object M of the category A and (cid:11) :
0 N
N=H ! N0=H0 is the homomorphism induced by (cid:11). e
Proof. The following commutative diagram of Lie algebras with exact columns
0 0 0 0
# # # #
!
(cid:1)(cid:1)(cid:1)! Ker(1F2(M) (cid:10)(cid:11)) !! Ker(1F1(M) (cid:10)(cid:11)) ! Ker(1M (cid:10)(cid:11)) ! Ker(cid:11)
!
e
# # # #
(cid:1)(cid:1)(cid:1)!! F2(M)(cid:10)N d10!!(cid:10)1N F1(M)(cid:10)N d00!(cid:10)1N M (cid:10)N !(cid:23) N=H ;
!
d11(cid:10)1N
1F2(M)(cid:10)(cid:11) # # 1F1(M)(cid:10)(cid:11) # 1M(cid:10)(cid:11) # (cid:11)e
(cid:1)(cid:1)(cid:1)!! F2(M)(cid:10)N0 d10!!(cid:10)1N0 F1(M)(cid:10)N0 d00(cid:10)!1N0 M (cid:10)N0 !(cid:23)0 N0=H0
!
d11(cid:10)1N0
# # # #
0 0 0 0
immediately induces the exactness of the sequence
(cid:1)(cid:1)(cid:1)(cid:0)! H (M;N0) (cid:0)(cid:14)!3 H (M;N;N0) (cid:0)j!2 H (M;N) (cid:0)i!2 H (M;N0) :
3 2 2 2
Using the "snake lemma" to the last two columns of this diagram one has the following
exact sequence
H (M;N) (cid:0)i!1 H (M;N0) (cid:0)(cid:14)!1 H (M;N;N0) (cid:0)j!0 H (M;N) (cid:0)i!0 H (M;N0) (cid:0)! 0 :
1 1 0 0 0
10 N.INASSARIDZE,E. KHMALADZEAND M. LADRA
We de(cid:12)ne the homomorphisms j and (cid:14) by
1 2
j (jxj) = (d0 (cid:10)1 )(x)
1 0 N
for x 2 fKer(1 (cid:10)(cid:11))\(d0 (cid:10)1 )(cid:0)1(Ker(1 (cid:10)(cid:11))\Ker(cid:23))g and
F1(M) 0 N M
(cid:14) (jyj) = j(d1 (cid:10)1 )(y0)(cid:0)(d1 (cid:10)1 )(y0)j
2 1 N 0 N
for y 2 Ker(d11 (cid:10)1N0)\Ker(d10 (cid:10)1N0), where y0 2 F2(M)(cid:10)N such that (1F2(M)(cid:10)(cid:11))(y0) =
y. It is easy to check that j and (cid:14) are well de(cid:12)ned and that the sequence (2) is exact in
1 2
terms H (M;N0), H (M;N;N0) and H (M;N) by virtue Proposition 3. (cid:3)
2 1 1
Remark. (a) If the actions of M and N satisfy the compatibility conditions (1) (in this
case M and N0 act on each other compatibly), then H (M;N;N0) = H (M;N00), where
0 0
N00 = Ker(cid:11).
(b)Let 0 ! (N00;0) ! (N;(cid:22)) ! (N0;(cid:23)) ! 0 be an exact sequence of crossed M-modules.
Thanks to the result in [14] there is a six-term exact non-abelian homology sequence
H (M;N00) (cid:0)! H (M;N) (cid:0)! H (M;N0)
(3) 1 1 1
(cid:0)! H (M;N0) (cid:0)! H (M;N) (cid:0)! H (M;N0) (cid:0)! 0 :
0 0 0
The (cid:12)rst (cid:12)ve terms of the sequence (3) coincide with the (cid:12)rst (cid:12)ve terms of the sequence
(2) and there is a natural homomorphism of (cid:3)-modules
H (M;N00) ! H (M;N;N0) :
1 1
Let
M
2
(4) D = # (cid:11)2
M !(cid:11)1 M
1
be a diagram in the category A with surjective (cid:11) . Let L (D;N) be the pullback of the
N 1 (cid:3)
induced diagram
F(cid:3)(M )(cid:10)A
2
# F(cid:3)((cid:11)2)(cid:10)1N
F(cid:3)(M )(cid:10)N F(cid:3)((cid:0)(cid:11)1!)(cid:10)1N F(cid:3)(M)(cid:10)N :
1
De(cid:12)ne H (D;N) = (cid:25) L (D;N), k (cid:21) 2.
k k(cid:0)1 (cid:3)
Theorem 8 (Mayer-Vietoris sequence). For any diagram (4) there is a long exact sequence
of (cid:3)-modules
(5) (cid:1)(cid:1)(cid:1) ! H (M;N) ! H (D;N) ! H (M ;N)(cid:8)H (M ;N) ! H (M;N)
k+1 k k 1 k 2 k
(cid:1)(cid:1)(cid:1) ! H (D;N) ! H (M ;N)(cid:8)H (M ;N) ! H (M;N) ! (cid:25) L (D;N)
2 2 1 2 2 2 0 (cid:3)
! (cid:25) (F(cid:3)(M )(cid:10)N)(cid:8)(cid:25) (F(cid:3)(M )(cid:10)N) ! (cid:25) (F(cid:3)(M)(cid:10)N) ! 0 :
0 1 0 2 0