Quadratic codimension growth and minimal varieties of unitary algebras with superinvolution

TL;DR

We address the problem of classifying unitary algebras with superinvolution according to the quadratic growth of their $*$-codimension sequence $c_n^*(A)$. The authors develop a cocharacter framework, identify explicit minimal varieties with quadratic growth, and prove a decomposition theorem that any unitary $*$-variety with quadratic growth is $T_*$-equivalent to a finite direct sum of algebras generating minimal varieties. The results yield a complete PI-equivalence classification for quadratic-growth unitary $*$-varieties and clarify the building-block algebras that control polynomial identities in algebras with superinvolution and graded involution contexts. These findings illuminate the structure of PI-theory in the graded-involution setting and provide tools for constructing and recognizing quadratic-growth varieties in practice.

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 $c_n^*({V}) \approx αn^k$ 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.

Theorems & Definitions (42)

• Example 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9
• Remark 2.10