THE G-GRADED IDENTITIES OF THE GRASSMANN ALGEBRA

Let G be a finite abelian group with identity element 1(G) and L = (g is an element of 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 broken vertical bar G'broken vertical bar dim(F) L(1)G' = Do and dimF L-g = infinity if g' 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 dim(F) L-g = infinity, where o(g) = 2, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of E in the case dim L(1)G Do and dim L9 < Do if g not equal 1(G).