A characterization of varieties of algebras of proper central exponent equal to two

TL;DR

This paper characterizes associative algebra varieties over a field of characteristic zero based on the exponential growth rate of their proper central polynomial sequences, specifically identifying those with growth equal to two.

Contribution

It provides a complete characterization of varieties with proper central polynomial growth exactly equal to two, extending previous exponential growth classifications.

Findings

Identifies varieties with exponential growth rate exactly two

Classifies varieties with growth greater than two

Builds on prior growth rate analyses

Abstract

Let $F$ be a field of characteristic zero and let $ \mathcal V$ be a variety of associative $F$-algebras. In \cite{regev2016} Regev introduced a numerical sequence measuring the growth of the proper central polynomials of a generating algebra of $ \mathcal V$. Such sequence $c_n^\delta(\mathcal V), \, n \ge 1,$ is called the sequence of proper central polynomials of $ \mathcal V$ and in \cite{GZ2018}, \cite{GZ2019} the authors computed its exponential growth. This is an invariant of the variety. They also showed that $c_n^\delta(\mathcal V)$ either grows exponentially or is polynomially bounded. The purpose of this paper is to characterize the varieties of associative algebras whose exponential growth of $c_n^\delta(\mathcal V)$ is greater than two. As a consequence, we find a characterization of the varieties whose corresponding exponential growth is equal to two.