Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups

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.

36 pages, 7 figures, v2: minor corrections, references added

References in corpus (4)

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• On the Yang-Baxter equation for the six-vertex model
• Q-operators in the six-vertex model
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• Integrable Matrix Models in Discrete Space-Time
• On non-abelian quadrirational Yang-Baxter maps
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Integrable quad equations derived from the quantum Yang-Baxter equation
• From Braces to Hecke algebras & Quantum Groups
• Quantum groups, Yang-Baxter maps and quasi-determinants
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• The Floquet Baxterisation
• The Coxeter relations and KP map for non-commuting symbols
• Anisotropic Landau-Lifshitz Model in Discrete Space-Time
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$
• Two-component Yang-Baxter maps and star-triangle relations
• Extensions of Yang-Baxter sets
• An algebraic approach to discrete time integrability
• Integrable multi-component difference systems of equations
• $2^n-$rational maps