Yang-Baxter maps: dynamical point of view

TL;DR

This paper reviews recent advances in the dynamical theory of Yang-Baxter maps, exploring their integrability and connections to various mathematical structures through concrete examples.

Contribution

It provides a comprehensive overview of the dynamical aspects of Yang-Baxter maps and their relations to integrability, matrix factorizations, and geometric structures.

Findings

Yang-Baxter maps exhibit integrable transfer dynamics

Connections to matrix KdV solitons and Poisson Lie groups are demonstrated

Concrete examples illustrate the interplay between algebraic and geometric aspects

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.