The second homology group of current Lie algebras

In view of the known relationship between central extensions and the second (co)homology group with coefficients in the trivial module, one of the main results of this paper can be considered as a generalization of this fact for general current Lie algebras, i.e., Lie algebras of the form L ⊗ A, where L is a Lie algebra and A is associative commutative algebra, equipped with bracket [x ⊗ a, y ⊗ b] = [x, y] ⊗ ab.

Theorem 0.1. Let L be an arbitrary Lie algebra over a field K of characteristic p = 2 and A an associative commutative algebra with unit over K. Then there is an isomorphism of K-vector spaces:

where the mapping S 2 (A) → A induced by multiplication in A and T (A) = ab ∧ c + ca ∧ b + bc ∧ a | a, b, c ∈ A .

Here B(L) is the space of coinvariants of the L-action on S 2 (L), HC 1 (A) is the firstorder cyclic homology group of A, and ∧ 2 and S 2 denote the skew and symmetric products, respectively. Notice that in the case L = [L, L], the third and fourth terms in the right-hand side of (0.1) vanish.

Many particular cases of this theorem were proved by different authors previously. An exhaustive description of all previous works on this theme may be found in [H] and [S].

For the first time, a cohomology formula of the type (0.1) has appeared in [S], where Theorem 0.1 was proved assuming that L is 1-generated over an augmentation ideal of its enveloping algebra. A. Haddi [H] obtained a result similar to Theorem 0.1 in the case where K is a field of characteristic zero (however, it seems that his arguments work over any field of characteristic p = 2, 3).

Our method of proof differs from all previous ones and is based on the Hopf formula expressing H 2 (L) in terms of a presentation 0 → I → L(X) → L → 0, where L = L(X) is the free Lie algebra over K freely generated by the set X:

(see, for example, [KS]). The contents of the paper are as follows. §1 is devoted to some technical preliminary results. In §2 we determine the presentation of a current Lie algebra L ⊗ A. In §3 Theorem 0.1 is proved. As it corollary we get in §4 a description of the space B(L ⊗ A). In §5 a “noncommutative version” of Theorem 0.1 is proved (Theorem 5.1). Namely, we derive the formula for the second homology group of the Lie algebra (A ⊗ B) (-) , where A, B are associative (noncommutative) algebras with unit, and (-) in superscript denotes passing to the associated Lie algebra. The technique used here is no longer based on the Hopf formula, but on more or less direct computations in some factorspaces of cycles. However, arguments used in proof, resemble, to a great extent, the previous ones. Getting a particular case B = M n (K), we recover, after a slight modification, an isomorphism H 2 (sl n (A)) ≃ HC 1 (A) obtained in [KL].

The following notational convention will be used: the letters a, b, c, . . . , possibly with suband superscripts, denote elements of algebra A, while letters u, v, w, . . . denote elements of the free Lie algebra L(X) with the set of generators X = {x i }, if the otherwise is not stated. L n (X) denotes the nth term in the derived series of L(X). The arrows and ։ denote injection and surjection, respectively.

All other undefined notions and notation are standard, and may be found, for example, in [F] for Lie algebra (co)homology, and in [LQ] for cyclic homology. In some places we use diagram chasing and 3 × 3-Lemma without explicitly mentioning it.

Looking at formula (0.1), one can distinguish between the first two “principal” terms and other two “non-principal” ones. In order to simplify calculations, we will obtain a variant of the Hopf formula leading to the appearance of “principal” terms only, and then the general case will be derived.

Each nonperfect Lie algebra L, i.e., not coinciding with its commutant [L, L], possesses a “trivial” homology classes of 2-cycles with coefficients in the module K, namely, classes whose representatives do not lie in L ∧ [L, L]. More precisely, consider a natural homomorphism ψ :

where π is induced by multiplication in L.

Proof. This is just an obvious consequence of a 5-term exact sequence derived from the Hochschild-Serre spectral sequence

Further, we need a version of Hopf formula for

and

Now consider an action of a Lie algebra L on S 2 (L) via

] be the space of coinvariants of this action. The dual B(L) * is the space of symmetric bilinear invariant forms on L.

Let I, J be ideals of L. Define B(I, J) to be the space of coinvariants of the action of L on I ∨ J. One has a natural embedding B(I, J) → B(L). The natural map L ∨ J → (L/I) ∨ ((I + J)/I) defines a surjection B(L, J) → B(L/I, (I + J)/I).

Lemma 1.3. The short sequence

Ker(B(L, J) → B(L/I, (I + J)/I))

Remark. Actually we need the following two cases of this Lemma:

(1

) and we get a short exact sequence

(2) I = [L, L] and J = L. Then taking into account that for an abelian Lie algebra M, B(M) ≃ S 2 (M), the short exact sequence (1.2) beco

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