The Magic Star of Exceptional Periodicity

TL;DR

This paper introduces an infinite family of algebraic structures extending exceptional objects like e8 and the exceptional Jordan algebra, revealing new connections and generalizations in algebra and mathematical physics.

Contribution

It constructs an infinite series of algebraic structures, including generalizations of e8 and the exceptional Jordan algebra, linking them through the concept of Magic Star algebras.

Findings

e8 is part of an infinite family of Magic Star algebras

The exceptional Jordan algebra is included in an infinite family of matrix algebras

The structures resemble lattice vertex algebras

Abstract

We present a periodic infinite chain of finite generalisations of the exceptional structures, including e8, the exceptional Jordan algebra (and pair), and the octonions. We demonstrate that the exceptional Jordan algebra is part of an infinite family of finite-dimensional matrix algebras (corresponding to a particular class of cubic Vinberg's T-algebras). Correspondingly, we prove that e8 is part of an infinite family of algebras (dubbed "Magic Star" algebras) that resemble lattice vertex algebras.