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!