Isotopes of Hurwitz algebras

TL;DR

This paper investigates isotopic algebras related to Hurwitz algebras, providing a classification framework and geometric insights, with applications to finite-dimensional composition algebras.

Contribution

It offers a comprehensive classification of isotopes of Hurwitz algebras, especially Hamilton's quaternions, using group actions and geometric descriptions.

Findings

Isomorphism classes correspond to group action orbits.

Complete geometric description of isotopes of Hamilton's quaternions.

Applications to classification of finite-dimensional composition algebras.

Abstract

We study the class of all algebras that are isotopic to a Hurwitz algebra. Isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. A complete, geometrically intuitive description of the category of isotopes of Hamilton's quaternions is given. As an application, we demonstrate how some results concerning the classification of finite-dimensional composition algebras can be deduced from our general results.