 By D.B. Fuks

There isn't any query that the cohomology of countless­ dimensional Lie algebras merits a short and separate mono­ graph. This topic isn't really cover~d by means of any of the culture­ al branches of arithmetic and is characterised by way of relative­ ly simple proofs and sundry software. furthermore, the subject material is greatly scattered in a number of study papers or exists purely in verbal shape. the speculation of infinite-dimensional Lie algebras differs markedly from the speculation of finite-dimensional Lie algebras in that the latter possesses robust type theo­ rems, which generally permit one to "recognize" any finite­ dimensional Lie algebra (over the sector of complicated or genuine numbers), i.e., locate it in a few checklist. There are classifica­ tion theorems within the thought of infinite-dimensional Lie al­ gebras in addition, yet they're weighted down via powerful restric­ tions of a technical personality. those theorems are helpful in most cases simply because they yield a substantial provide of curiosity­ ing examples. we commence with a listing of such examples, and extra direct our major efforts to their study.

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Additional info for Cohomology of Infinite-Dimensional Lie Algebras

Sample text

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

C-1 (g; A) = 0, we have HO (g; A) ° Ker [do: Co (g; A) = A _ C (g; A)J {a EO A I ga = for all g EO g} = InvgA. = = [In an equally obvious way one can construct the isomorphism Ho (g; A) = 2. ] Since the differential do: CO (g) _C1(g) is trivial, we have H1 (g) ° Ker [d 1: C1 (g) = g' _ CZ (g)J = {y EO g' I y ([g, hI) = for all g, h EO g} = (g/[g, gl)'. ) 3. The space H1 (g; g) can be interpreted as the space of "exterior derivations" of the algebra g. Recall that the homomorphism rp: 9 ~ Ii is said to be a derivation ifrp ([g, hI) = [rp (g), hI g>-+ [go, gl, + [g, where rp (h)l; go -= g as examples note "inner derivations" is a fixed element.

For m= 0 it is finite dimensional, while for is infinite dimensional; for n bra form Wm , = 0 m> 0 it it becomes the Lie alge- Its elements may be written "in coordinates" in the CHAPTER 1 54 where fl' gj are power series in Xl> • • • , whose coeffi- Xm , cients are elements of the Grassmann algebra in Yl, ... , Yn' When taking the commutator of such "vector fields," one should keep in mind that signs should be changed in changing places of the YI with each other or with 818Yi a iJyz (Yl/\ Y2) =- (for example, Yl)' If the field (1) belongs to W (m, n)p, then its diver- gence is, by definition, the expression ~ (8fJ8xJ 8Yj); (-1)P~ (8g/ by linearity the definition of divergence can be car- ried over to the entire space Fields with trivial W(m, n).