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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**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).