Partially Alternative Real Division Algebras With A Few Imaginary Units

TL;DR

This paper classifies four-dimensional partially alternative real division algebras with multiple imaginary units, extending classical theorems and revealing infinitely many distinct classes with specific automorphism groups.

Contribution

It provides a complete classification of certain four-dimensional partially alternative real division algebras, generalizing classical algebraic theorems and characterizing their automorphism groups.

Findings

Infinitely many isomorphism classes of such algebras

Automorphism group is SO(3) or Z_2 depending on the algebra

Generalization of Frobenius and Hurwitz theorems in this context

Abstract

In this paper, we extend the investigation of four-dimensional partially alternative algebras $\mathcal A$ initiated in \cite{HNT}. The partial alternativity condition, a natural generalization of the alternativity axiom, broadens the class of alternative algebras, enabling the exploration of non-associative structures with weakened algebraic constraints. We focus on four-dimensional partially alternative real division algebras (RDAs) equipped with at least three distinct imaginary units and a reflection. Under these constraints, we achieve a complete classification of the isomorphism classes of such algebras and find that there are, in fact, infinitely many distinct classes. This classification generalizes the classical Frobenius and Hurwitz theorems in the four-dimensional setting. Moreover, we explicitly characterize automorphism groups for these algebras showing that it is either $SO(3)$ if $\mathcal A\cong \mathbb H$ or $\mathbb Z_2$, otherwise.