By D.B. Fuks
There isn't any query that the cohomology of endless dimensional Lie algebras merits a short and separate mono graph. This topic isn't cover~d by way of any of the culture al branches of arithmetic and is characterised by means of relative ly uncomplicated proofs and sundry software. additionally, the subject material is greatly scattered in a number of study papers or exists merely in verbal shape. the idea of infinite-dimensional Lie algebras differs markedly from the idea of finite-dimensional Lie algebras in that the latter possesses robust class theo rems, which generally permit one to "recognize" any finite dimensional Lie algebra (over the sector of advanced or actual numbers), i.e., locate it in a few record. There are classifica tion theorems within the conception of infinite-dimensional Lie al gebras besides, yet they're laden via powerful restric tions of a technical personality. those theorems are helpful in general simply because they yield a substantial offer of curiosity ing examples. we commence with an inventory of such examples, and additional direct our major efforts to their study.
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A class of infinitesimal deformations of the algebra g. Are the infinitesimal deformations of this class derivatives with respect to some parameter of true deformations of the algebra g? 2G), they are not in general; now we shall indicate some necessary condition for this to take place. Suppose ~ is a cocycle which represents a. Assume that the infinitesimal deformation TJ corresponds to a true deformation h: 9 ~< 9 X IR - g, and expand h in a Taylor series in t: Further, for the commutator [gl' g2lt = h (gl' g2' t) write out the Jacobi identity and set the coefficients of the Taylor series of its left-hand side equal to zero: + ...
Poincare duality. Suppose the Lie algebra g is fi- nite dimensional and dim g =- n_ Then dim Cn (g) = 1 and for any nonzero f E Cn (g) the formula a, b ....... ab (f) def ines a nondegenerate pairing (7) and, together with it, an isomorphism C~ (9) ~ (C n-" (g»' = Cn- k (9). (8) In order that this isomorphism be compatible with the differ- a, entials d and it is necessary that the algebra 9 possess a supplementary property: it must be unitary. , / is a cycle. The latter For the algebra Ie g, determined by means of the "structural constants" e,j for some basis el, ...
3). 2°. Suppose C = {C q , tF: complex over K and w: g' _ C Cq _C q + 1 } is a multiplicative is an arbitrary linear map. 1 Then there exists a unique homogeneous multiplicative homomorphism Q: W (g) _ C, whose restriction to Alg' = g' C W (g) coincides with tion Q I Slg - 6l w (cp), w. The construction is obvious: maps cP E Slg' = g' where of the cochain into the difference A 2 w (d(p) is the differential dcp E A2q' = C2 (g') (P E g' = Cl(g) the restric- in the complex C· (g). For example, if in the principal G - bundle E - X are given the connection wE we G is a Lie group, Ql (E; g) ( g the corresponding Lie algebra), then, by assigning to each functional cp E g' the form cp ° wE we obtain a linear Q, map g' _ QIL; in view of the above, this map can be canonically extended to a homomorphism W (g) _ Q' (E), which is an important invariant of the connection; in particular, it may be used to define characteristic classes of fiber bundles.