Varieties of group-graded algebras of proper central exponent greater than two

TL;DR

This paper characterizes associative G-graded algebra varieties with a proper central G-exponent greater than two, using a finite list of algebras and growth rate analysis of central G-polynomials.

Contribution

It provides a finite classification of varieties with proper central G-exponent greater than two, linking algebraic structure to polynomial growth rates.

Findings

A finite list of G-graded algebras characterizes varieties with exponent > 2.

The exponent > 2 if and only if the variety contains at least one algebra from the list.

Characterization of varieties with exponent exactly 2 based on growth properties.

Abstract

Let $F$ be a field of characteristic zero and let $ \mathcal V $ be a variety of associative $F$-algebras graded by a finite abelian group $G$. To a variety $ \mathcal V $ is associated a numerical sequence called the sequence of proper central $G$-codimensions, $c^{G,\delta}_n(\mathcal V), \, n \ge 1.$ Here $c^{G,\delta}_n(\mathcal V)$ is the dimension of the space of multilinear proper central $G$-polynomials in $n$ fixed variables of any algebra $A$ generating the variety $\mathcal V.$ Such sequence gives information on the growth of the proper central $G$-polynomials of $A$ and in \cite{LMR} it was proved that $exp^{G,\delta}(\mathcal V)=\lim_{n\to\infty}\sqrt[n]{c_n^{G,\delta}(\mathcal V)}$ exists and is an integer called the proper central $G$-exponent. The aim of this paper is to characterize the varieties of associative $G$-graded algebras of proper central $G$-exponent greater than two. To this end we construct a finite list of $G$-graded algebras and we prove that $exp^{G,\delta}(\mathcal V) >2$ if and only if at least one of the algebras belongs to $\mathcal V$. Matching this result with the characterization of the varieties of almost polynomial growth given in \cite{GLP}, we obtain a characterization of the varieties of proper central $G$-exponent equal to two.