A structural classification of algebras with graded involution and quadratic codimension growth

TL;DR

This paper classifies associative G-graded algebras with graded involution that have quadratic codimension growth, linking their structure to minimal varieties and cocharacter multiplicities.

Contribution

It provides a complete classification of such algebras up to equivalence, connecting algebra structure with codimension growth and minimal varieties.

Findings

Complete classification of algebras with quadratic codimension growth

Establishment of correspondence between minimal varieties and cocharacter multiplicities

Every variety with at most quadratic growth is generated by a decomposable algebra

Abstract

The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a fundamental numerical invariant, as it captures the growth rate of the polynomial identities of a given algebra. Special partial classification results have been obtained, with particular interest devoted to algebras equipped with additional structure. In this paper, we consider associative G-graded algebras endowed with a graded involution. We provide a complete classification, up to equivalence, of unitary algebras with quadratic codimension growth. Our approach establishes a direct correspondence between the algebras generating minimal varieties and the nonzero multiplicities appearing in the decomposition of the proper cocharacters. As a consequence, we establish that every variety with at most quadratic growth is generated by an algebra that decomposes as a direct sum of algebras generating minimal varieties of at most quadratic growth.