On a class of local Lie algebras over a manifold

This letter presents a study of the automorphisms and the derivations of a large class of local Lie algebras over a manifold M (in the sense of Shiga and Kirillov) called Lie algebras of order O over M. It is shown that, in general, the algebraic structure of such an algebra GAMMA characterizes the differentiable structure of M and that the Lie algebra of derivations of GAMMA is a Lie algebra of differential operators of order 1 over M obtained in a natural way as the space of sections of a vector bundle canonically associated to GAMMA. (Auth.)