Varieties of graded $W$-algebras and asymptotic behavior of codimension growth

TL;DR

This paper develops a theory of graded polynomial identities for finite-dimensional $G$-graded algebras, computes identities for specific cases, and explores varieties with near-polynomial codimension growth.

Contribution

It introduces a generalized theory of graded identities, proves the existence of the graded exponent, and explicitly computes identities for certain graded matrix algebras.

Findings

The graded generalized exponent exists and equals the ordinary exponent.

Explicit identities are computed for the $UT_2$ algebra with $bZ_2$-grading.

Examples of varieties with almost polynomial codimension growth are provided.

Abstract

Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the graded multiplier algebra of $A.$ In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the $G$-graded generalized identities of $UT_2,$ the $2 \times 2$ upper triangular matrix algebra equipped with its canonical $\mathbb{Z}_2$-grading, under all the possible graded $W$-actions. Finally, we exhibit examples of varieties of graded $W$-algebras with almost polynomial growth of the codimensions.