On the Construction of Trigonometric Solutions of the Yang-Baxter Equation

TL;DR

This paper presents a generalized method for constructing trigonometric R-matrices for tensor products of irreducible quantum algebra representations, leading to many new solutions of the Yang-Baxter equation.

Contribution

It extends the tensor product graph method to different representations, enabling the systematic construction of new trigonometric R-matrices.

Findings

Decomposition of R-matrices into projection operators.

Construction of many new trigonometric R-matrices.

Generalization of the tensor product graph method.

Abstract

We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two irreducible representations of a quantum algebra $U_q(\G)$. Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation) are constructed using this approach.