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

Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (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...

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Bibliographic Details
Main Authors Lecomte, P. B. A, Ovsienko, V. Yu
Format Journal Article
LanguageEnglish
Published 11.05.1999
Subjects
Mathematics - Differential Geometry
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Summary:Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (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 $\cal S$. This cohomology space is closely related to the $Vect(M)$-modules, ${\cal D}_\lambda(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $\lambda$.
Bibliography:CPT-99/P.3816
DOI:10.48550/arxiv.math/9905058