Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Abstract

LetM be a smooth manifold,S the space of polynomial on ¢bers functions on TM (i.e., of symmetric contravariant tensor ¢elds).We compute the ¢rst cohomology space of the Lie algebra, VectOMU, of vector ¢elds on M with coef¢cients in the space of linear differential operators onS. This cohomology space is closely related to the VectOMU-modules,DlOMU ,o f linear differential operators on the space of tensor densities on M of degree l.