On nonassociative graded-simple algebras over the field of real numbers

TL;DR

This paper extends the loop algebra construction to classify graded-simple nonassociative algebras over the real numbers, including alternative and Jordan types, providing a comprehensive classification framework.

Contribution

It introduces a generalized loop algebra approach for graded-simple algebras over real fields and classifies specific types of nonassociative graded algebras.

Findings

Classification of graded-simple alternative algebras over reals

Classification of graded-simple Jordan algebras of degree 2

Classification of graded-division alternative algebras

Abstract

We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence.