Star-Varieties of proper central exponent greater than two

TL;DR

This paper investigates associative algebras with involution that have a proper central *-exponent greater than two, identifying specific algebras that characterize such varieties.

Contribution

It constructs a finite list of algebras with involution and proves their relevance to varieties with exponent exceeding two.

Findings

Established the existence of the proper central *-exponent as an integer.

Identified a finite list of algebras associated with exponent greater than two.

Proved that such varieties must contain at least one algebra from this list.

Abstract

Let $F$ be a field of characteristic zero and let $ \mathcal V^* $ be a variety of associative $F$-algebras with involution *. Associated to $ \mathcal V^* $ are three sequences: the sequence of \(*\)-codimensions \( c^{*}_n(\mathcal V^*) \), the sequence of central \(*\)-codimensions \( c^{*,z}_n(\mathcal V^*) \) and the sequence of proper central \(*\)-codimensions \( c^{*,\delta}_n(\mathcal V^*) \). These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomial and the proper central *-polynomial of any generating algebra with involution $A$ of $ \mathcal V^*.$ In \cite{MR2022} it was proved that $exp^{*,\delta}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,\delta}(\mathcal V^*)}$ exists and is an integer called the proper central $*$-exponent. The aim of this paper is to study the varieties of associative algebras with involution of proper central $*$-exponent greater than two. To this end we construct a finite list of algebras with involution and we prove that if $exp^{*,\delta}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$.