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

Abstract

Yang–Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang–Baxter equation. For a set X = Ω × V, where V is a vector space and Ω is regarded as a space of parameters, a linear parametric YB map is a YB map Y: X × X → X × X such that Y is linear with respect to V and one has πY = π for the projection π: X × X → Ω × Ω. These conditions are equivalent to certain parametric nonlinear algebraic relations for the components of Y. Such a map Y may be nonlinear with respect to parameters from Ω. We present general results on such maps, including the 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 and Mikhailov 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.