Semigroup graded algebras and codimension growth of graded polynomial identities

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.

21 pages; Minor misprints are corrected

Cited by in corpus (5)

• Semigroup graded algebras and graded PI-exponent
• On $H$-simple not necessarily associative algebras
• Actions of Ore extensions and growth of polynomial $H$-identities
• Graded group actions and generalized $H$-actions compatible with gradings
• (Co)module algebras and their generalizations