Daniela La Mattina

Almost polynomial growth: Classifying varieties of graded algebras

Let G be a finite group, V a variety of associative G-graded algebras and cnG(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.

On algebras of polynomial codimension growth

Let A be an associative algebra over a field F of characteristic zero and let

Varieties of Algebras with Superinvolution of Almost Polynomial Growth

c_n(A), n=1, 2, ldots

Polynomial codimension growth of algebras with involutions and superinvolutions

cn(A),n=1,2,…, be the sequence of codimensions of A. It is well-known that

Minimal star-varieties of polynomial growth and bounded colength

c_n(A), n=1, 2, ldots

On the graded identities and cocharacters of the algebra of 3×3 matrices☆

cn(A),n=1,2,…, cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.