Quadratic codimension growth and minimal varieties of unitary algebras with superinvolution

TL;DR

This paper classifies minimal varieties of unitary superinvolution algebras with quadratic codimension growth, providing a structural characterization and showing their generation by direct sums of minimal varieties.

Contribution

It offers a classification and structural description of minimal varieties with quadratic codimension growth in algebras with superinvolution.

Findings

Quadratic codimension growth varieties are generated by direct sums of minimal varieties.

Structural characterization of all such algebras up to PI-equivalence.

Complete classification of minimal varieties with quadratic growth.

Abstract

Let $A$ be an associative algebra with a superinvolution $*$ over a field of characteristic zero, and let $c_n^*(A)$, $n = 1, 2, \ldots$, denote its sequence of $*$-codimensions. It is well known that this sequence is either polynomially bounded or grows exponentially. In the polynomial case, a central problem in PI-theory is the classification of varieties ${V}$ for which $c_n^*({V}) \approx \alpha n^k$ for a given $k$. One of the main objectives of this paper is to classify minimal varieties of unitary algebras endowed with a superinvolution that exhibit quadratic codimension growth. We obtain a structural characterization, up to PI-equivalence, of all unitary algebras with quadratic codimension growth. As a consequence, we show that any unitary variety of quadratic codimension growth is generated by a direct sum of algebras generating minimal varieties.