On the classification of G-graded twisted algebras

TL;DR

This paper classifies G-graded twisted algebras over commutative rings, focusing on associative cases over complex and real numbers, and nonassociative cases with symmetry conditions for finite cyclic groups, using group cohomology.

Contribution

It provides a complete classification of G-graded twisted algebras, extending known results to nonassociative cases with specific symmetry and group conditions.

Findings

Full classification of associative G-graded twisted algebras over complex and real fields.

Extension of classification to nonassociative cases with symmetry conditions.

Application of group cohomology methods to algebra classification.

Abstract

Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each summand W_{g} is a free rank one R -module, and W has no monomial zero divisors (for each pair of nonzero elements w_{a},w_{b} en W_{a} and W_{b} their product is not zero, w_{a}w_{b}\neq 0). It is also assumed that W has an identity element. In this article, methods of group cohomology are used to study the general problem of classification under graded isomorphisms. We give a full description of these algebras in the associative cases, for complex and real algebras. In the nonassociative case, an analogous result is obtained under a symmetry condition of the corresponding associative function of the algebra, and when the group providing the grading is finite cyclic.