Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems

TL;DR

This paper explores the structure and construction of Yang--Baxter maps, especially linear parametric ones, using Darboux transformations and refactorisation techniques, introducing new maps with nonlinear parameter dependence.

Contribution

It provides a comprehensive analysis of algebraic relations defining Yang--Baxter maps and introduces methods to generate new maps from known ones, including linear approximations from Darboux transformations.

Findings

New linear parametric Yang--Baxter maps are constructed.

Methods to derive maps from Darboux transformations are developed.

Maps with nonlinear parameter dependence are presented.

Abstract

Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vector space and $\Omega$ is regarded as a space of parameters, a linear parametric YB map is a YB map $Y\colon X\times X\to X\times X$ such that $Y$ is linear with respect to $V$ and one has $\pi Y=\pi$ for the projection $\pi\colon X\times X\to\Omega\times\Omega$. These conditions are equivalent to certain nonlinear algebraic relations for the components of $Y$. Such a map $Y$ may be nonlinear with respect to parameters from $\Omega$. We present general results on such maps, including clarification of the structure of the algebraic relations that define them and several transformations which allow one to obtain new such maps from known ones. Also, methods for constructing such maps are described. In particular, developing an idea from [Konstantinou-Rizos S and Mikhailov A V 2013 J. Phys. A: Math. Theor. 46 425201], we demonstrate how to obtain linear parametric YB maps from nonlinear Darboux transformations of some Lax operators using linear approximations of matrix refactorisation problems corresponding to Darboux matrices. New linear parametric YB maps with nonlinear dependence on parameters are presented.