# 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.