2025/10/27 by Lucio Centrone, Plamen Koshlukov, Centrone, Lucio +3
#math.RA
paper · pdf · doi:10.48550/arxiv.2510.23869
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∈ G Ag, 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 the choice of g and h but not on the elements ag and ah. It is immediate that β is a bicharacter on G. The regular decomposition above is minimal if for every g∈ G with β(g,h)=β(g,k) one has h=k. In this paper we prove that if G=ℤ2 then every G-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of ℤ2-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated graded subalgebras of a ℤ2-graded regular algebra having a minimal regular decomposition.