Triple Products and Yang-Baxter Equation (I): Octonionic and Quaternionic Triple Systems

TL;DR

This paper explores how the Yang-Baxter equation can be expressed as a triple product equation and finds new rational solutions using octonionic and quaternionic triple systems.

Contribution

It introduces a novel approach to solving the Yang-Baxter equation through algebraic relations in octonionic and quaternionic triple systems.

Findings

Derived new rational solutions to the Yang-Baxter equation

Established algebraic relations for octonionic and quaternionic systems

Demonstrated the applicability of triple product formulations

Abstract

We can recast the Yang-Baxter equation as a triple product equation. Assuming the triple product to satisfy some algebraic relations, we can find new solutions of the Yang-Baxter equation. This program has been completed here for the simplest triple systems which we call octonionic and quaternionic. The solutions are of rational type.