Group graded algebras and varieties with quadratic codimension growth

TL;DR

This work addresses the problem of classifying unitary $G$-graded algebras with polynomial codimension growth, focusing on the quadratic-growth case. It develops the $G$-graded invariant framework (identities, codimensions, and proper cocharacters) and leverages symmetric-group representation theory to analyze multiplicities, culminating in an explicit building-block description. The main result shows that any unitary $G$-graded variety with quadratic growth is $T_G$-equivalent to a finite direct sum of algebras that generate minimal $G$-graded varieties, with a precise list of such algebras and their interrelations. This provides a concrete, modular structure for quadratic-growth graded PI-varieties and extends the PI-theory to the setting of finite-group gradings, offering a clear path to constructing and understanding these varieties via their minimal components.

Abstract

Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial identities satisfied by $A$. It is known that this sequence is either polynomially bounded or grows exponentially. In this paper, we study unitary $G$-graded varieties of polynomial codimension growth. In particular, we classify the varieties generated by unitary algebras with quadratic codimension growth and show that these varieties can be described as a direct sums of algebras that generate minimal $G$-graded varieties.

Theorems & Definitions (40)

• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• Remark 3.2
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• proof