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

By D.B. Fuks

There is not any query that the cohomology of endless­ dimensional Lie algebras merits a quick and separate mono­ graph. This topic isn't cover~d by way of any of the culture­ al branches of arithmetic and is characterised via relative­ ly easy proofs and sundry software. furthermore, the subject material is greatly scattered in a variety of examine papers or exists in basic terms 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 category theo­ rems, which generally let one to "recognize" any finite­ dimensional Lie algebra (over the sector of complicated or actual numbers), i.e., locate it in a few record. There are classifica­ tion theorems within the conception of infinite-dimensional Lie al­ gebras to boot, yet they're laden by way of powerful restric­ tions of a technical personality. those theorems are invaluable commonly simply because they yield a substantial offer of curiosity­ ing examples. we commence with an inventory of such examples, and additional direct our major efforts to their study.

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Gp), to the image of this homomorphism being contained in Hom (AP (9/6),- A) C Hom (AP g, A). V (g/6), A), which is obviously an epimorphism with kernel FP+ICP+q (g; A). Thus we get the isomorphism This isomorphism commutes with the differentials: for c E FP(;p+q (g; A), hi, ... , hq+l EO ~, gI, . . •• gp) P + 8=1 ~ ~ (- 1)Ht e (ItI' t=1 ... e 8=1 = ~ I<;;8

1 •... Ar ) (g; A)}. 2b. The inclusion C;o ... O) (g; A) -C· (g; A) induces an isomorphism in cohomology. 2b can be stated and proved just as their cohomology duals except, and this is an important difference, that the algebra 9 and the module A no longer need be assumed topological and must possess not a topological but a real basis, constituted by eigenvectors of all the transformations g ....... [gi, gl 3. and a ........ gia. The Laplace operator. The following is a consider- ably simplified finite-dimensional analog of the Hodgede Rham theory.

G (gl ... g,) = possesses another 9 -module -gl ... g. g, supplies the spaces Cq (g; U (g» and the second structure with 9 -module structures such that the differentials aq : Cq (g; U (g» _ Cq _ 1 (g; U (g» turn out to be g-homomorphisms. Having this structure in mind, we shall view C as a complex of 9 -modules and g -homomorphisms. Obviously for any g-moduleA , the spaces cq(g; GENERAL THEORY 23 are none other than Homu(S) (C q (g; U (g», A), A (59U(9) Cq (g; U (g», lIomu(S) ([0,,: Cq (g; U (g)) ~ Cq-dg; U (g))], it!

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