Non-commutative rational Yang-Baxter maps

TL;DR

This paper constructs non-commutative solutions to the Yang-Baxter equation derived from lattice systems, extending known integrable models like sine-Gordon and Boussinesq to a non-commutative, non-autonomous setting.

Contribution

It introduces a novel approach to non-commutative Yang-Baxter maps based on multidimensional consistency and centrality conditions, extending integrable lattice equations.

Findings

Derived non-commutative Yang-Baxter maps from lattice systems.

Recovered non-commutative sine-Gordon equation.

Presented a non-autonomous non-commutative Boussinesq equation.

Abstract

Starting from multidimensional consistency of non-commutative lattice modified Gel'fand-Dikii systems we present the corresponding solutions of the functional (set-theoretic) Yang-Baxter equation, which are non-commutative versions of the maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel'fand-Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular we recover known non-commutative version of Hirota's lattice sine-Gordon equation, and we present an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.