Semigroup graded algebras and codimension growth of graded polynomial identities

TL;DR

This paper constructs examples of semigroup graded algebras with non-integer codimension growth exponents, and identifies conditions under which the graded PI-exponent is an integer, advancing understanding of polynomial identities in graded algebras.

Contribution

It demonstrates the existence of finite dimensional semigroup graded algebras with non-integer PI-exponents and establishes conditions for the graded PI-exponent to be an integer.

Findings

Existence of semigroup graded algebras with non-integer PI-exponent

Conditions for integer graded PI-exponent in unital or cancellative cases

Equality of ordinary and graded PI-exponents in certain cases

Abstract

We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-graded algebra is unital, or $T$ is a cancellative semigroup, then the $T$-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.