Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory

TL;DR

This paper explores functional relations in solvable lattice models, linking transfer matrices, quantum group modules, and the thermodynamic Bethe ansatz, proposing new relations for models tied to all simple Lie algebras.

Contribution

It clarifies the role of exact sequences in quantum group modules and proposes universal functional relations for models associated with all simple Lie algebras.

Findings

Solutions of the functional relations also solve thermodynamic Bethe ansatz equations at high temperature.

Functional relations satisfy constraints from the fusion procedure.

Proposed relations are consistent with known structures and constraints.

Abstract

We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for $sl(r+1)$ models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.