Classification of division gradings on finite-dimensional simple real algebras

TL;DR

This paper classifies division gradings by abelian groups on finite-dimensional simple real algebras, providing a comprehensive understanding of their structure and equivalence classes.

Contribution

It offers the first complete classification of division gradings on simple real algebras, linking linear algebra over the field of two elements to the problem.

Findings

Classification up to isomorphism and equivalence achieved

Division gradings determine all gradings on simple real algebras

Linear algebra over GF(2) is key to the proofs

Abstract

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give the classification, up to isomorphism, of (not necessarily division) gradings on such algebras. Linear algebra over the field of two elements plays an interesting role in the proofs.