Yang-Baxter maps: dynamical point of view

Abstract

A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.

24 pages. Extended version of lectures given at the meeting "Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004)

References in corpus (6)

• The A^{(1)}_M automata related to crystals of symmetric tensors
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Soliton Cellular Automata Associated With Crystal Bases
• Non-unitary set-theoretical solutions to the Quantum Yang-Baxter Equation
• Tropical R and Tau Functions
• Poisson Lie groups and Hamiltonian theory of the Yang-Baxter maps

Cited by in corpus (29)

• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• On Quadrirational Yang-Baxter Maps
• Simplex and Polygon Equations
• Grassmann extensions of Yang-Baxter maps
• Yang-Baxter maps associated to elliptic curves
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Some properties of the k-dimensional Lyness' map
• Poisson structures for lifts and periodic reductions of integrable lattice equations
• Relativistic Collisions as Yang-Baxter maps
• Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups
• The Floquet Baxterisation
• Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Yang-Baxter Maps from the Discrete BKP Equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Quantum spaces associated to multipermutation solutions of level two
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlevé Equations
• Multipermutation solutions of the Yang--Baxter equation
• Geometric lifting of the integrable cellular automata with periodic boundary conditions
• Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map
• Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
• Garside structure on monoids with quadratic square-free relations
• Two FRT bialgebroids and their relations
• FRT Construction for Dynamical Yang-Baxter Maps
• Differential equations for the closed geometric crystal chains