Graded polynomial identities, group actions, and exponential growth of Lie algebras

Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsurʼs conjecture on asymptotic behavior for codimensions of polynomial G-identities of L. As a consequence, we prove the analog of Amitsurʼs conjecture for graded codimensions of any finite dimensional Lie algebra graded by a finite Abelian group.