By D.B. Fuks

There isn't any query that the cohomology of limitless­ dimensional Lie algebras merits a short and separate mono­ graph. This topic isn't really cover~d through any of the culture­ al branches of arithmetic and is characterised via relative­ ly user-friendly proofs and sundry software. in addition, the subject material is commonly scattered in a variety of examine papers or exists merely 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 strong category 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 record. There are classifica­ tion theorems within the idea of infinite-dimensional Lie al­ gebras in addition, yet they're laden via powerful restric­ tions of a technical personality. those theorems are important customarily simply because they yield a substantial provide of curiosity­ ing examples. we start with an inventory of such examples, and additional direct our major efforts to their study.

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Gr) @ (hi /\ ••• /\ hp+-+- ~ a ® g~ ... g~ ... g~-r ® (g~ /\ gl/\ ... /\ gr) ® (hI /\ ... /\ hp +q_ r) j=l determines a homotopy in the complex E~. which joins 0 with Thus, p id. q = 0 = E;'q for p =F O. At the same time, it is obvious that Eg,q = Eg,q (0) A), C~ (I); tfo"l = O. ';,'1' The homology part of the theorem is proved; the proof of its cohomology part is similar. §6. LIE SUPERALGEBRAS 1. Main definitions. " This passage consists in assuming that all the objects under consideration possess a l2-grading, in writing minus one to the appropriate power in the appropriate places of all definitions, and adding the prefix "super" to all the terms.

The Laplace operator. The following is a consider- ably simplified finite-dimensional analog of the Hodgede Rham theory. Assume that the Lie algebra 9 is finite-dimensional or graded in such a way that all the spaces C~)(g) are finite dimensional (for example, this property is possessed by all gradings of the form 9 = ~AEzng(A) with finite-dimensional g(A), for which there exists an N, such that ·9(ka..... k n ) = 0 whenever 47 GENERAL THEORY min (kl' ... , kn ) < -N ). in every space chosen. l (g) -+- Cr~/ The (g), while the operator ator which commutes with d and = C~A) (g).

Using cohomology language, we can state many other algebraic problems related to Lie algebras. Without going into detail (and in places without even giving exact definitions) we shall list certain problems of this type in this subsection. Everywhere we assume that a homomorphism f of the Lie algebra g into the Lie algebra A. Suppose 1'] E C~ (g; g), ~ E mal deformations of the algebras g, homology classes of the cocycles ~. is given. C2 (6; 6) are Infinites i- 6, and a, p are the co- 1'],~. Question: does there GENERAL THEORY 39 exist an infinitesimal deformation (of the homomorphism f) which extends the infinitesimal deformations many such deformations exist?

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