Solutions to the constant Yang-Baxter equation: additive charge conservation in three dimensions

TL;DR

This paper classifies all solutions to the constant Yang-Baxter equation in three dimensions under an additive charge-conservation ansatz, revealing a unique family of solutions with four parameters and analyzing their symmetry structures.

Contribution

It provides a complete classification of solutions to the constant Yang-Baxter equation with additive charge conservation in three dimensions, including a detailed analysis of their symmetry algebra.

Findings

Single 3-parameter family of solutions in generic 3D case

Complete structure of the symmetry algebra for these solutions

Solutions for cases with fewer nonzero sector-coupling parameters

Abstract

We find all solutions to the constant Yang--Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ in three dimensions, subject to an additive charge-conservation ansatz. This ansatz is a generalisation of (strict) charge-conservation, for which a complete classification in all dimensions was recently obtained. Additive charge-conservation introduces additional sector-coupling parameters -- in 3 dimensions there are $4$ such parameters. In the generic dimension 3 case, in which all of the $4$ parameters are nonzero, we find there is a single 3 parameter family of solutions. We give a complete analysis of this solution, giving the structure of the centraliser (symmetry) algebra in all orders. We also solve the remaining cases with three, two, or one nonzero sector-coupling parameter(s).