Integrable Chiral Potts Model and the Odd-Even Problem in Quantum Groups at Roots of Unity

TL;DR

This paper explores the relationship between the integrable chiral Potts model and quantum groups at roots of unity, highlighting how gauge choices affect quantum group constructions and addressing the odd-even problem.

Contribution

It introduces two distinct quantum group constructions arising from gauge choices in the six-vertex model, explaining their applicability to odd and even N, and generalizes to sl(m,n) models.

Findings

Different gauge choices lead to distinct quantum group structures.

One construction is effective only for odd N, the other for all N.

Generalization to sl(m,n) vertex models is discussed.

Abstract

At roots of unity the $N$-state integrable chiral Potts model and the six-vertex model descend from each other with the $\tau_2$ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different $q$-Pochhammer symbols, one construction only working well for $N$ odd, the other equally well for all $N$. We also address the generalization based on the sl$(m,n)$ vertex model.