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

TL;DR

This paper classifies G-graded twisted algebras over finite abelian groups using group cohomology, focusing on associative cases and those with specific symmetry conditions, over real or complex fields.

Contribution

It provides a comprehensive classification of associative G-graded twisted algebras and extends previous work to include algebras with certain symmetry properties.

Findings

Classified all associative G-graded twisted algebras over finite abelian groups.

Extended classification to include algebras with symmetry conditions.

Utilized group cohomology methods for the classification.

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 a direct sum over the elements of G of one dimensional K-vector spaces. It is also assumed that W has no monomial which is a zero divisor. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and the field K is either the real numbers or the complex numbers. 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 (Velez et. al., 2014) we present a classification of all G-graded twisted algebras that satisfy certain symmetry condition.