Graded group actions and generalized $H$-actions compatible with gradings
A. S. Gordienko
TL;DR
The paper develops a unified framework for graded group actions on graded algebras via graded pseudoautomorphisms and generalized $H$-actions compatible with gradings. It establishes that the associated graded automorphism groups form algebraic objects, and proves the radical is invariant and a graded version of Wedderburn–Artin decompositions holds under these actions, enabling invariant structure theory. It extends Amitsur-type codimension growth results to graded $G$- and $H$-identities, showing that for finite-support gradings these graded theories reduce to the classical $\mathbbm{k}^T\otimes H$ setting and that the PI-exponent exists in this broad context. The work includes explicit structural descriptions for key algebras (e.g., $M_{1,1}(\mathbbm{k})$, Grassmann-type algebras) and provides a coherent route from generalized actions to graded identities, with implications for codimension growth and invariant decompositions in a wide class of algebras.
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 $G$.