Group graded algebras and varieties with quadratic codimension growth

TL;DR

This paper classifies unitary G-graded varieties with quadratic codimension growth, showing they are direct sums of algebras generating minimal G-graded varieties, thus advancing understanding of polynomial identity growth in graded algebras.

Contribution

It provides a classification of G-graded varieties with quadratic codimension growth and describes their structure as direct sums of minimal G-graded varieties.

Findings

Classified varieties generated by unitary algebras with quadratic G-graded codimension growth.

Showed these varieties are direct sums of algebras generating minimal G-graded varieties.

Established that G-graded codimension growth is either polynomial or exponential.

Abstract

Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial identities satisfied by $A$. It is known that this sequence is either polynomially bounded or grows exponentially. In this paper, we study unitary $G$-graded varieties of polynomial codimension growth. In particular, we classify the varieties generated by unitary algebras with quadratic codimension growth and show that these varieties can be described as a direct sums of algebras that generate minimal $G$-graded varieties.