Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions

A. S. Khoroshkin

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 19, Tome 335 (2006), pp. 205-230

Résumé

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