On the Classification of G-Graded Twisted Algebras over Finite Abelian Groups

Abstract

Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as W = ⊕g∈GWg with WaW b ⊂ W ab , and where each Wg is a one dimensional K-vector space. It is also assumed that W has no monomial which is a zero divisor which means that for each pair of nonzero elements wa ∈ Wa, w b ∈ W b , wa • w b = 0. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and K = C or K = R. In this article, using methods of group cohomology, we classify all associative G-graded twisted algebras in the case G is a finite abelian group. On the other hand, by generalizing some of the arguments developed in [1] we present a classification of all G-graded twisted algebras that satisfy certain symmetry condition.