Graded commutative algebras: examples, classification, open problems

TL;DR

This paper explores $ ext{G}$-graded commutative algebras, providing examples, a classification result, and posing an open problem to advance understanding of their structure and properties.

Contribution

It offers a unifying perspective on $ ext{G}$-graded commutative algebras, including examples like quaternions and Clifford algebras, along with a recent classification and an open problem.

Findings

Classification of $ ext{G}$-graded commutative algebras

Examples including quaternions and Clifford algebras

Formulation of an open problem in the field

Abstract

We consider $\G$-graded commutative algebras, where $\G$ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on the subject. We then give a recent classification result and formulate an open problem.