Classification of the four-dimensional power-commutative real division algebras

TL;DR

This paper provides a complete classification of four-dimensional power-commutative real division algebras, showing they are isotopes of quadratic division algebras and characterizing those with a unique non-zero idempotent.

Contribution

It introduces a novel classification framework linking four-dimensional power-commutative division algebras to quadratic division algebras via isotopes.

Findings

Every four-dimensional power-commutative real division algebra is an isotope of a quadratic division algebra.

Complete classification of four-dimensional power-commutative real division algebras.

Characterization of finite-dimensional power-commutative real division algebras with a unique non-zero idempotent.

Abstract

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimension four and eight is reduced to the description of quadratic division algebras. In dimension four this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterised.