Lie derivations on rings of differential operators

Derivations on rings of differential operators are studied. In particular, we ask whether Lie derivations are forced to be associative derivations. This is established for the Weyl algebras, which provides the details of a theorem of A. Joseph.

The ideas are extended to localizations of Weyl algebras. As a corollary, the implication is verified for the universal enveloping algebras of nilpotent Lie algebras.