The G-graded Identities of the Grassmann Algebra

Let G be a finite abelian group with identity element 1G and L = ⊕ g∈G L g be an infinite dimensional G-homogeneous vector space over a field of characteristic 0. Let E = E(L) be the Grassmann algebra generated by L. It follows that E is a G-graded algebra. Let |G| be odd, then we prove that in order to describe any ideal of G-graded identities of E it is sufficient to deal with G′-grading, where |G′| ≤ |G|, dimF L1G′ =∞ and dimF Lg ′ <∞ if g′ 6 = 1G′ . In the same spirit of the case |G| odd, if |G| is even it is sufficient to study only those G-gradings such that dimF Lg =∞, where o(g) = 2, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of E in the case dimL1G =∞ and dimLg <∞ if g 6= 1G.