A structural classification of algebras with graded involution and quadratic codimension growth
Wesley Quaresma Cota, Luiz Henrique de Souza Matos, Ana Cristina Vieira
TL;DR
The paper provides a complete classification, up to $T_G^*$-equivalence, of unitary $(G,*)$-algebras with quadratic codimension growth, by linking the nonzero multiplicities in proper $(n_1,...,n_{2k})$-cocharacters to minimal varieties. It develops a framework based on $G$-graded algebras with graded involution, cocharacter theory, and the notion of proper $(G,*)$-polynomials to identify building blocks that generate minimal quadratic-growth varieties. A rich catalog of explicit algebras (e.g., $C_m^g$, $C_{m,*}^g$, $U_{3,*}^g$, $\mathcal{G}_{2,*}^{g,h}$, $W_{ u_j}^{p,q}$, etc.) is provided with their codimension growth and cocharacter data, forming the foundational ingredients. The main result shows that any unitary variety of quadratic growth is a direct sum of algebras generating minimal varieties, enabling a finite decomposition into well-understood building blocks and advancing the PI-theory of graded-involution algebras.
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.