On infinite dimensional algebras with regular gradings

Let G be a finite abelian group and let K be an algebraically closed field of characteristic 0. We consider associative unital algebras A over K graded by G, that is A=⊕g∈GAg, where the vector subspaces Ag satisfy AgAh⊆Ag+h for every g, h∈G. Such a G-grading is called regular whenever for every n-tuple (g1,…,gn)∈Gn there exist homogeneous elements ai∈Agi such that a1⋯an≠0 in A; furthermore, for every g, h∈G and every ag∈Ag, ah∈Ah one has agah=β(g,h)ahag for some β(g,h)∈K∗. Here β(g,h) depends only on g and h but not on the elements ag and ah. It is immediate that β is a skew-symmetric bicharacter on G. The regular decomposition above is minimal if whenever β(g,h)=β(g,k) for every g∈G, then h=k. In this paper we characterize the generators of the graded variety generated by the natural Z2-grading on the Grassmann algebra in terms of Z2-graded regular algebras with minimal regular decomposition. Furthermore we describe the finitely generated graded subalgebras of a Z2-graded regular algebra having a minimal regular decomposition. We recall that regular gradings and the corresponding decompositions play an important role in the description of numerical invariants of PI algebras as proved in the papers [1, 4, 6, 26].