Star-Varieties of proper central exponent greater than two
F. S. Benanti, A. Valenti
TL;DR
This work characterizes varieties of associative algebras with involution by their proper central $*$-exponent. It introduces a finite obstruction set of $14$ specific involutive algebras $\mathcal A_i$, computes their exponents, and proves that any $*$-variety with $exp^{*,\delta}>2$ must contain one of these algebras. Consequently, it classifies minimal $*$-varieties with central exponent $3$ or $4$, showing non-comparability among them and clarifying how Grassmann envelopes and finite-dimensional simple superalgebras govern the growth of central and proper central $*$-codimensions. The results extend the PI-algebra framework to algebras with involution and provide a concrete obstruction-based dichotomy for the rich growth behavior in this setting.
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 \( c^{*,δ}_n(\mathcal V^*) \). 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 $exp^{*,δ}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,δ}(\mathcal V^*)}$ 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 $exp^{*,δ}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$.