Determinants in Jordan matrix algebras

TL;DR

This paper defines and explores a determinant concept within Jordan matrix algebras, specifically for hermitian matrices over biquaternions and complex octonions, aiding in understanding their algebraic structure.

Contribution

It introduces a natural determinant notion in matrix JB*-algebras and analyzes properties relevant to the structure of Cartan factors of type 6.

Findings

Defined determinants for hermitian matrices over biquaternions and octonions

Characterized minimal projections in Cartan factor of type 6

Described automorphisms of the Cartan factor of type 6

Abstract

We introduce a natural notion of determinant in matrix JB$^*$-algebras, i.e., for hermitian matrices of biquaternions and for hermitian $3\times 3$ matrices of complex octonions. We establish several properties of these determinants which are useful to understand the structure of the Cartan factor of type $6$. As a tool we provide an explicit description of minimal projections in the Cartan factor of type $6$ and a variety of its automorphisms.