ar X iv : m at h / 03 12 16 2 v 1 [ m at h . D G ] 8 D ec 2 00 3 Derivations of the Lie algebras of differential operators ∗

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) = P ol(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 DR (M) ⊕ R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.