Identities in the smash product of the universal envelope of a Lie superalgebra and a group algebra

Let $H$ be a Hopf algebra and $A$ an $H$-module algebra. Then one can form the smash product $A\#H$, which is a generalization of the ordinary tensor product (the latter occurs if the action of $H$ on $A$ is trivial). The case when $A\#H$ satisfies a polynomial identity is studied. Appropriate delta sets are introduced and necessary conditions on the action of $H$ on $A$ in terms of these delta sets for a certain class of algebras are given. The main theorem treats the special case when $H$ is a group algebra acting on a Lie superalgebra $L$ of characteristic zero. In this case the results obtained on delta sets, in combination with known facts about group algebras and universal enveloping algebras, enable one to give necessary and sufficient conditions for the existence of a polynomial identity in $U(L)\#H$.