Derivations of the Lie algebras of differential operators

Abstract

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 D1(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, H1(D(M), D(M)) = HDR1(M), H1 (D1(M), D1(M)) = HDR1(M) ⊕ R2, and H1(S(M), S(M)) = HDR1 (M) ⊕ R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.