Solving the n-color ice model

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.