Derivations of the Lie Algebras of Differential Operators

Details Derivations of the Lie Algebras of Differential Operators Derivations of the Lie Algebras of Differential Operators

2003-12-08

arxiv.org/abs/math/0312162v2

Indag. Mathem., N.S., 16 (2), 181-200, 2005

Mathematics

Differential Geometry

LaTeX, 15 pages

Scientific paper

This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M, of its Lie subalgebra D^1(M) of all linear first-order differential operators of M, and of the Poisson algebra S(M)=Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in D(M). It turns out that, in terms of the Chevalley cohomology, H^1(D(M),D(M))=H^1_{DR}(M), H^1(D^1(M),D^1(M))=H^1_{DR}(M)\oplus\R^2, and H^1(S(M),S(M))=H^1_{DR}(M)\oplus\R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.

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