Classification of some graded not necessarily associative division algebras I

TL;DR

This paper classifies certain finite-group graded real division algebras, including classical and novel non-associative types, expanding understanding of their algebraic structures.

Contribution

It provides a classification of graded real division algebras with specific basis and scalar conditions, identifying both known and new non-associative division algebras.

Findings

Classified classical algebras: complex, quaternion, octonion

Discovered new non-associative division algebras

Focused on algebras graded by abelian groups of order ≤8

Abstract

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group $G$, and have a basis $\{v_g|g\in G\}$ as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in $\{1,-1\}$. We classify here those graded by an abelian group $G$ of order $|G|\leq 8$ with $G$ non--isomorphic to $\z/8\z$. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non--associative division algebras.