On locally complex algebras and low-dimensional Cayley-Dickson algebras

TL;DR

This paper revisits classical theorems on real division algebras, introduces locally complex algebras, and extends these theorems to include the fifth Cayley-Dickson algebra, the sedenions.

Contribution

It introduces the concept of locally complex algebras and extends Frobenius' and Zorn's theorems to include the sedenions.

Findings

Classifies locally complex algebras of dimension up to 4

Provides short proofs of Frobenius and Zorn theorems

Extends classical theorems to include the sedenions

Abstract

The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.