Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups

TL;DR

This paper develops a framework linking quantum groups, Yang-Baxter maps, and integrable discrete equations, providing a method to construct and analyze quantum and classical integrable systems on lattices.

Contribution

It introduces a general scheme connecting quantized Lie algebras with integrable discrete systems via Yang-Baxter maps, including their classical limits.

Findings

Constructed integrable quantum evolution systems on quadrilateral lattices.

Derived classical Yang-Baxter maps and Hamiltonian systems from quantum models.

Applied the framework explicitly to the algebra U_q(sl(2)) leading to discrete Liouville equations.

Abstract

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra $U_q(sl(2))$ leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.