Minimal varieties of algebras with graded involution and quadratic growth

TL;DR

This paper investigates the asymptotic codimension growth of finite-dimensional algebras with graded involution, classifying minimal varieties with quadratic growth within the framework of graded algebras.

Contribution

It provides a classification of minimal varieties generated by finite-dimensional graded involution algebras exhibiting quadratic growth.

Findings

Classified minimal varieties with quadratic growth for $(G,*)$-algebras.

Analyzed asymptotic behavior of codimension sequences.

Extended understanding of polynomial growth in graded involution algebras.

Abstract

Subalgebras of upper triangular matrix algebras have played a fundamental role in the classification of minimal varieties of polynomial growth. Such classification has become a source of study in recent years since it leads to the more general classification of varieties of polynomial growth $n^k$, as has already been proven in many contexts for several values of $k$. In this paper, we study the asymptotic behavior of the sequence of codimensions of algebras graded by a finite group $G$ and endowed with a graded involution $*$, also called $(G,*)$-algebras. We classify the minimal varieties generated by a finite-dimensional $(G,*)$-algebra with quadratic growth.