A New Solution to the Star--Triangle Equation Based on U$_q$(sl(2)) at Roots of Unit

TL;DR

This paper introduces new solutions to the Yang--Baxter equation using quantum group intertwiners, leading to novel lattice models, scattering matrices, and potential knot invariants, especially at roots of unity.

Contribution

It presents a new approach to solving the Yang--Baxter equation via semi-cyclic representations of $U_q(sl(2))$ at roots of unity, connecting to lattice models and knot theory.

Findings

New solutions to the Yang--Baxter equation using quantum group intertwiners.

Construction of Boltzmann weights for lattice models similar to the chiral Potts model.

Infinite-dimensional braid group representations at $N oinity$ suggesting knot invariants.

Abstract

We find new solutions to the Yang--Baxter equation in terms of the intertwiner matrix for semi-cyclic representations of the quantum group $U_q(s\ell(2))$ with $q= e^{2\pi i/N}$. These intertwiners serve to define the Boltzmann weights of a lattice model, which shares some similarities with the chiral Potts model. An alternative interpretation of these Boltzmann weights is as scattering matrices of solitonic structures whose kinematics is entirely governed by the quantum group. Finally, we consider the limit $N\to\infty$ where we find an infinite--dimensional representation of the braid group, which may give rise to an invariant of knots and links.