From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

TL;DR

This paper derives explicit finite-dimensional solutions to the Yang-Baxter equation by restricting known infinite-dimensional solutions on principal series representations, using fusion procedures and revealing simple formulas.

Contribution

It introduces a method to obtain finite-dimensional Yang-Baxter solutions from principal series representations, unifying rational and trigonometric cases with explicit formulas.

Findings

Explicit formulas for finite-dimensional solutions of the Yang-Baxter equation.

Demonstration of the fusion procedure aligning with restriction methods.

Unified approach for rational and trigonometric solutions.

Abstract

We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group $\mathrm{SL}(2,\mathbb{C})$ or Faddeev's modular double. Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces. In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches.