Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold

Abstract

Let $M$ be a smooth manifold, $\mathcal{S}$ the space of polynomial on fibers functions on $T^{\ast }M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on $\mathcal{S}$. This cohomology space is closely related to the $Vect(M)$-modules, ${\mathcal{D}}_{\lambda }(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $\lambda$.

14 pages, LaTeX