Solving the n-color ice model

TL;DR

This paper derives explicit algebraic conditions and parametrizations for solutions to the Yang-Baxter equation in n-color lattice models, generalizing known models like the six-vertex and including quantum affine algebra solutions.

Contribution

It provides explicit algebraic conditions and a parametrization for solutions to the Yang-Baxter equation in n-color lattice models, extending classical results.

Findings

Derived algebraic conditions for solvability of n-color models.

Provided explicit parametrization of all solutions.

Connected solutions to quantum affine Lie algebra representations.

Abstract

Given an arbitrary choice of two sets of nonzero Boltzmann weights for $n$-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the Yang-Baxter equation. Furthermore we provide an explicit one-dimensional parametrization of all solutions in this case. These $n$-color lattice models are so named because their admissible vertices have adjacent edges labeled by one of $n$ colors with additional restrictions. The two-colored case specializes to the six-vertex model, in which case our results recover the familiar quadric condition of Baxter for solvability. The general $n$-color case includes important solutions to the Yang-Baxter equation like the evaluation modules for the quantum affine Lie algebra $U_q(\hat{\mathfrak{sl}}_n)$. Finally, we demonstrate the invariance of this class of solutions under natural transformations, including those associated with Drinfeld twisting.