Graded group actions and generalized $H$-actions compatible with gradings

TL;DR

This paper introduces graded group actions on algebras via graded pseudoautomorphisms, generalizing super- and pseudoinvolutions, and explores their structural implications including radical stability and polynomial identity growth.

Contribution

It defines graded pseudoautomorphisms, proves radical stability under these actions, and establishes invariant versions of classical theorems and conjectures for graded algebras.

Findings

Jacobson radical is stable under graded pseudoautomorphisms

Invariant Wedderburn-Artin theorem for graded algebras

Analog of Amitsur's conjecture for graded polynomial identities

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 $G$.