Cohomology of Infinite-Dimensional Lie Algebras by D.B. Fuks

By D.B. Fuks

There isn't any query that the cohomology of endless­ dimensional Lie algebras merits a short and separate mono­ graph. This topic isn't really cover~d via any of the culture­ al branches of arithmetic and is characterised via relative­ ly easy proofs and sundry program. additionally, the subject material is largely scattered in numerous study papers or exists simply in verbal shape. the idea of infinite-dimensional Lie algebras differs markedly from the idea of finite-dimensional Lie algebras in that the latter possesses strong type theo­ rems, which typically enable one to "recognize" any finite­ dimensional Lie algebra (over the sphere of complicated or actual numbers), i.e., locate it in a few checklist. There are classifica­ tion theorems within the concept of infinite-dimensional Lie al­ gebras to boot, yet they're weighted down via robust restric­ tions of a technical personality. those theorems are precious more often than not simply because they yield a substantial offer of curiosity­ ing examples. we start with a listing of such examples, and additional direct our major efforts to their study.

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Simplest general properties. sequences and induced homomorphisms. A. Coefficient For any short exact sequence of g-modules we have the following exact coefficient sequences: ... 1 2 ) . . -> Hq (g; -> 11'1 (g; A 3 ) -> Hq+l (g; i'h) .. 1 2 ) -> llq (g; A 3 ) - . H q_1 (g; Al ) ••• Their construction is standard. : Hq (g; A) - 27 Hq (~; A)" (here A is a g-module in which I introduces a g-module structure). B. Poincare duality. Suppose the Lie algebra g is fi- nite dimensional and dim g =- n_ Then dim Cn (g) = 1 and for any nonzero f E Cn (g) the formula a, b .......

4). is The formula q>, ljJ -- ~ K «p (t), ljJ' (t)) dt 8' determines a cocycle in C2 ((g8')P01), which is not cohomologous to zero, and the cohomology class of this cocycle generates H2 ((gS')Pol). 3). GENERAL THEORY 35 The space H2 (g; 7" 9) can be interpreted as the set of classes of infinitesimal deformations of the algebra g. Recall that a deformation of the algebra 9 is a smooth map h: 9 X g X IR - g. such that h (glt g2' 0) =[gl. g21 t E IR the opera- and for every tion determines a Lie algebra structure in h.

Gl' ... gs '=1 1}"I-1 [D(AI (e)] Ug" gil, g~, ... l, ... , gq) A =1= 0) a homotopy joining the iden- is (for tity of the complex ... ) (g) with the zero map. Let us indicate two generalizations of this theorem. First, if the g·-module A possesses a topological basis constituted by the eigenvectors of the transformation then, by letting A(A) ==, {a E AI a ...... 7 of the complex C· (g; A). 2, we can prove the following statement. 2a. The inclusion C;o) (g; A) _C" (g; A) in- duces an isomorphism in homology.

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