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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Extra info for Cohomology of Infinite-Dimensional Lie Algebras
Sample text
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.
