Graded polynomial identities and codimensions: Computing the exponential growth

摘要

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→∞) 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.