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

By D.B. Fuks

There is not any query that the cohomology of countless­ dimensional Lie algebras merits a short and separate mono­ graph. This topic isn't cover~d through any of the culture­ al branches of arithmetic and is characterised through relative­ ly undemanding proofs and sundry software. in addition, the subject material is generally scattered in numerous study papers or exists purely in verbal shape. the idea of infinite-dimensional Lie algebras differs markedly from the speculation of finite-dimensional Lie algebras in that the latter possesses robust class theo­ rems, which typically let one to "recognize" any finite­ dimensional Lie algebra (over the sphere of complicated or genuine numbers), i.e., locate it in a few checklist. There are classifica­ tion theorems within the thought of infinite-dimensional Lie al­ gebras besides, yet they're weighted down via powerful restric­ tions of a technical personality. those theorems are worthwhile ordinarily 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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It < ... < +q- jq-l), 1} and Obviously, for all a E CI' (9; g), b E Cq (g; g) we have [a, bJ = - (-l)(P-l) (q-O [b, a]. A short direct computation shows that for all a b E Cq (g; g), c E Cr (g; :g) we have = (3) E:: CP (g; g), 20 CHAPTER 1 (-1)(P-l) (q-l) [[a, b), eJ + (_1)(T-l)(P-l) + (_1)(q-l) (T-l) [[e, aJ, bJ = [[b, e], aJ 0, (4) and a longer, but equally direct one, shows that d [a, bJ = fda, bJ - It is clear from (5) that if (_1)P-l [a, dbJ. (5) a, bare cocycles, then [a, bJ is alsu a cocycle, and the cohomology class of the latter is determined by the cohomology classes of a and b.

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 ....... ab (f) def ines a nondegenerate pairing (7) and, together with it, an isomorphism C~ (9) ~ (C n-" (g»' = Cn- k (9). (8) In order that this isomorphism be compatible with the differ- a, entials d and it is necessary that the algebra 9 possess a supplementary property: it must be unitary.

3. Every element of the space be represented by a unique harmonic cocycle from H~) (g) can CrA) (g). In other words, there is a natural isomorphism Proof. ) C;A) (g)}. ) (9) is acyclic and the differential of this complex is trivial. But we already know the second assertion, while the first follows from the fact that the homomorphisms + B: Lol)CfA) (g) -.. ) (9) of the complex 4. °) CHAPTER 1 with the trivial map. Relationship with induced and co induced modules. This relationship is expressed by the following theorem.

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