Abstract
Let M be a smooth manifold, \({\mathcal{S}} \) 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 \({\mathcal{S}} \). This cohomology space is closely related to the Vect(M)-modules, \({\mathcal{D}}\) λ(M), of linear differential operators on the space of tensor densities on M of degree λ.
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Lecomte, P.B.A., Ovsienko, V.Y. Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold. Compositio Mathematica 124, 95–110 (2000). https://doi.org/10.1023/A:1002447724679
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DOI: https://doi.org/10.1023/A:1002447724679