Poisson identities of enveloping algebras

Abstract.

Let L be a restricted Lie algebra. The symmetric algebra S _p (L) of the restricted enveloping algebra u(L) has the structure of a Poisson algebra. We give necessary and sufficient conditions on L in order for the symmetric algebra S _p (L) to satisfy a multilinear Poisson identity. We also settle the same problem for the symmetric algebra S(L) of a Lie algebra L over an arbitrary field.