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)

Cited by in corpus (9)

• Simplex and Polygon Equations
• Yang-Baxter maps associated to elliptic curves
• Poisson structures for lifts and periodic reductions of integrable lattice equations
• Quantum spaces associated to multipermutation solutions of level two
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps
• Multipermutation solutions of the Yang--Baxter equation
• Garside structure on monoids with quadratic square-free relations