LIE ALGEBRA OF FORMAL VECTOR FIELDS WHICH ARE EXTENDED BY FORMAL g-VALUED FUNCTIONS

摘要

In this paper, we consider the infinite-dimensional Lie algebra W_n ∝ g directX O_n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl_n ⊕g) by the (2n + 1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n ∝ g directX O_n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles.