Simple graded commutative algebras

TL;DR

This paper classifies simple finite-dimensional associative graded commutative algebras over real and complex numbers, showing they are Clifford algebras, and explores extensions to non-associative cases.

Contribution

It proves Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over  or , and introduces non-associative extensions.

Findings

Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over  and .

The paper extends the concept to non-associative graded commutative algebras.

Provides a classification framework for graded commutative algebras.

Abstract

We study the notion of $\Gamma$-graded commutative algebra for an arbitrary abelian group $\Gamma$. The main examples are the Clifford algebras already treated by Albuquerque and Majid. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over $\mathbb{R}$ or $\mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.