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

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By D.B. Fuks

There isn't any query that the cohomology of endless­ dimensional Lie algebras merits a quick and separate mono­ graph. This topic isn't really cover~d via any of the culture­ al branches of arithmetic and is characterised by means of relative­ ly basic proofs and sundry program. additionally, the subject material is commonly scattered in quite a few study papers or exists in basic terms in verbal shape. the speculation of infinite-dimensional Lie algebras differs markedly from the idea of finite-dimensional Lie algebras in that the latter possesses robust type theo­ rems, which generally enable one to "recognize" any finite­ dimensional Lie algebra (over the sector of advanced or actual numbers), i.e., locate it in a few checklist. There are classifica­ tion theorems within the idea of infinite-dimensional Lie al­ gebras to boot, yet they're laden by way of robust restric­ tions of a technical personality. those theorems are priceless ordinarily simply because they yield a substantial provide of curiosity­ ing examples. we commence with a listing of such examples, and additional direct our major efforts to their study.

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1. Let Obviously, C" (g; A) = PC" (g; A) :::=> ••• :::=> F"C" (g; A):::=> pr+1C" (g; A) = 0, and definitions imply that dFPCp+q (g; A) C FPCp+qH (g; A). Thus, {PP} is a filtration in the complex C' (g; A). corresponding spectral sequence is precisely The {E~,q. d~,IJ}. 42 CHAPTER 1 Let us compute the initial terms. \"b ~ lIum (AI'g, A), into the homomorphism ... ;\/i'l g, A .. /\gp >-+ sending ~ Itd\ c (hi"'" h,/. and the inclusion cEO FPCI>+'1 (g; A) is equivalent Ifl' .... gp), to the image of this homomorphism being contained in Hom (AP (9/6),- A) C Hom (AP g, A).

At the same time, elements of the image of the differential are inner derivations: do: Co (g; g) _ Cl (g; g) Co (g; g) gE 9 = we have do g (h) = - 4. for gh = [-g, hI. d ~ (1) ~:d 0->9- gl->IK-'>O. To the cohomology class of the cocycle e E CI (9; g) corresponds the class of the extension O-> 9 g t- (g, 0) v gEB", (g, A) I-A IV 0 '",- , where the Lie algebra structure in 9 EB K is defined by the formula The Jacobi identity for this commutator is equivalent to c being a cocycle: the left-hand side of this identity for (6'f, AI), (g2' "'2)' (g3' A3), after regrouping terms, becomes + [[g2' g31.

A). A». Hom (AP (g/6). /V(g/~), A) -+Hom (AP+l (g/~), A) exists. Of course, we do have the differential d: CP+l (g, ~; CP (g, ~; A)-+ which is a natural homomorphism A), Homo (AP (g/~), A) ~ Hom~ (AP+l (g/~), A). and an obvious verification shows that the diagram Hq (~) ® CP (g, ~; A) ~ Hq (~) ® CP+l (g, ~; A) II II Hq (~; Hom~ (AP (g/~), A» ~ H q (~; Hom (AP (g/~), A» (1) Hq (~; Hom, (AP+l (g/~), A» ~ -=-. Hq (~; Hom (AP+l (g/~). A)), d P• q whose vertical arrows are induced by the inclusions Homb (AT (g/~), A) -+ Hom (A r (g/~), A), is commutative.

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