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

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, $c_n^{G,\delta }(\mathcal{V}),n\ge 1.$ Here $c_n^{G,\delta }(\mathcal{V})$ 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 $ex{p}^{G,\delta }(\mathcal{V})={\lim }_{n\to \infty }\sqrt[n]{c_n^{G,\delta }(\mathcal{V})}$ 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 $ex{p}^{G,\delta }(\mathcal{V})>2$ 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.