3D compatible ternary systems and Yang-Baxter maps

TL;DR

This paper explores the relationship between 3D compatible systems, quasigroups, and Yang-Baxter maps, introducing new multi-parametric examples and methods for deriving these maps from higher-dimensional systems.

Contribution

It establishes a connection between 3D compatibility of equations on quad-graphs and the construction of parametric Yang-Baxter maps, providing new examples and reduction techniques.

Findings

New multi-parametric Yang-Baxter maps derived from quasigroup structures

Identification of conditions under which dynamical YB maps are parametric

Reduction of higher-dimensional maps to parametric YB maps

Abstract

According to Shibukawa, ternary systems defined on quasigroups and satisfying certain conditions provide a way of constructing dynamical Yang-Baxter maps. After noticing that these conditions can be interpreted as 3-dimensional compatibility of equations on quad-graphs, we investigate when the associated dynamical Yang-Baxter maps are in fact parametric Yang-Baxter maps. In some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints. Conversely, parametric YB maps on quasigroups with an invariance condition give rise to 3-dimensional compatible systems. The application of this method on spaces with certain quasigroup structures provides new examples of multi-parametric YB maps and 3-dimensional compatible systems.