Universal R-matrix and functional relations

TL;DR

This paper systematically reviews the use of quantum groups in deriving functional relations for integrable systems, exemplified by the six-vertex model, and proves these relations independently of specific representations.

Contribution

It provides a unified framework for functional relations in integrable models using quantum groups, with a detailed case study of the six-vertex model.

Findings

Established representation-independent functional relations

Applied the framework specifically to the six-vertex model

Proved the full set of functional relations in this context

Abstract

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.