Almost polynomial growth: Classifying varieties of graded algebras

Abstract

Let G be a finite group, V a variety of associative G-graded algebras and cn G(V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the G-graded varieties of almost polynomial growth by giving a complete list of finite-dimensional G-graded algebras generating them.