Graded polynomial identities and codimensions: computing the exponential growth

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 $c_n^G(A), \ n=1,2, \ldots,$ the sequence of graded codimensions of $A$ and we prove that if $A$ satisfies an ordinary polynomial identity, $\lim_{n\to \infty}\sqrt[n]{c_n^G(A)}$ 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 $G\times \mathbb{Z}_2$-graded algebra related to $A$.