Graded polynomial identities and codimensions: Computing the exponential growth

Abstract

Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -graded algebra related to A.