Jordan algebras, exceptional groups, and higher composition laws

TL;DR

This paper explores the connection between Jordan algebras, exceptional groups, and higher composition laws, introducing an integral construction and an algorithmic approach to study orbit spaces, leading to new examples and potential extensions.

Contribution

It presents an integral version of the Freudenthal construction linking Jordan algebras and exceptional groups, and proposes an algorithmic method to identify new spaces with composition laws.

Findings

Discovered two new orbit space examples with composition laws

Developed an algorithmic approach for studying orbit spaces

Identified several potential spaces for future composition law applications

Abstract

We consider an integral version of the Freudenthal construction relating Jordan algebras and exceptional algebraic groups. We show how this construction is related to higher composition laws of M.Bhargava in number theory. We propose an algorithmic approach to studying orbit spaces of groups underlying higher composition laws. Using this method we discover two new examples of spaces sharing similar properties, and indicate several more examples of spaces where such composition laws may be introduced.