$G$-identities of non-associative algebras

The main class of algebras considered in this paper is the class of algebras of Lie type. This class includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras, quantum Lie algebras, and many others. We prove that if a finite group $G$ acts on such an algebra $A$ by automorphisms and anti-automorphisms and $A$ satisfies an essential $G$-identity, then $A$ satisfies an ordinary identity of degree bounded by a function that depends on the degree of the original identity and the order of $G$. We show in the case of ordinary Lie algebras that if $L$ is a Lie algebra, a finite group $G$ acts on $L$ by automorphisms and anti-automorphisms, and the order of $G$ is coprime to the characteristic of the field, then the existence of an identity on skew-symmetric elements implies the existence of an identity on the whole of $L$, with the same kind of dependence between the degrees of the identities. Finally, we generalize Amitsur's theorem on polynomial identities in associative algebras with involution to the case of alternative algebras with involution.