Identities of graded algebras and codimension growth

Abstract

Let A = ⊕ g ∈ G A g A=\oplus _{g\in G}A_g be a G G -graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component A e A_e to that of the whole of A A , in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where A A is finite dimensional and A e A_e has polynomial growth.