Exotic matrix models: the Albert algebra and the spin factor

TL;DR

This paper explores matrix models associated with the Albert algebra and the spin factor, extending the study of Jordan algebra-based matrix models beyond the classical symmetric and Hermitian cases.

Contribution

It introduces and analyzes matrix models linked to the Albert algebra and the spin factor, which are the remaining simple formally real Jordan algebras not previously studied.

Findings

Developed matrix models for the Albert algebra and spin factor.

Extended understanding of Jordan algebra matrix models to new algebraic structures.

Provided foundational work for further exploration of exotic matrix models.

Abstract

The matrix models attached to real symmetric matrices and the complex/quaternionic Hermitian matrices have been studied by many authors. These models correspond to three of the simple formally real Jordan algebras over R. Such algebras were classified by Jordan, von Neumann, and Wigner in the 30s, and apart from these three there are two others: (i) the spin factor L_{1,n}, an algebra built on R^{n+1}, and (ii) the Albert algebra A of 3 by 3 Hermitian matrices over the octonions. In this paper we investigate the matrix models attached to these remaining cases.