Graded group actions and generalized H-actions compatible with gradings

Abstract

We introduce the notion of a graded group action on a graded algebra or, which is the same, a group action by graded pseudoautomorphisms. An algebra with such an action is a natural generalization of an algebra with a super- or a pseudoinvolution. We study groups of graded pseudoautomorphisms, show that the Jacobson radical of a group graded finite dimensional associative algebra A over a field of characteristic 0 is stable under graded pseudoautomorphisms, prove the invariant version of the Wedderburn–Artin Theorem and the analog of Amitsur's conjecture for the codimension growth of graded polynomial G-identities in such algebras A with a graded action of a group G.