Graded algebras with homogeneous involution and varieties of almost polynomial growth

TL;DR

This paper studies the growth of polynomial identities in graded algebras with homogeneous involution, providing classifications of varieties with almost polynomial growth based on the exclusion of certain algebras.

Contribution

It introduces the concept of homogeneous involution in graded algebras and classifies varieties with almost polynomial growth in this context.

Findings

Characterization of polynomial growth varieties via exclusion of specific algebras

Classification of varieties with almost polynomial growth

Analysis of codimension sequence behavior in graded algebras with involution

Abstract

An important aspect in the theory of algebras with polynomial identities is the study of the asymptotic behavior of the codimension sequence $c_n(A),\, n\geq 1,$ which measures the growth of polynomial identities of a given algebra $A$. In this context, graded identities naturally arise as prominent tools, since ordinary polynomial identities can be viewed as a particular case of graded identities. Moreover, as an involution does not necessarily preserve the homogeneous components of a grading, it is natural to consider the notion of a homogeneous involution. In this work, we investigate the behavior of the codimension sequence in the setting of $G$-graded algebras endowed with a homogeneous involution. More specifically, we characterize the varieties of polynomial growth in terms of the exclusion of a list of algebras from the variety. As a consequence, we provide the classification of the varieties with almost polynomial growth in this setting.