Reductions and degenerate limits of Yang-Baxter maps with $3\times 3$ Lax matrices

TL;DR

This paper extends a family of Yang-Baxter maps with 3x3 Lax matrices by adding parameters, explores their reductions and limits, and discusses their integrability, contributing to the classification of such maps.

Contribution

It introduces new parameterized Yang-Baxter maps with preserved Poisson structures and analyzes their reductions and degenerate limits, advancing the classification of these maps.

Findings

Derived several new birational Yang-Baxter maps.

Identified integrability properties of the new maps.

Explored reductions and limits of the maps.

Abstract

We generalise a family of quadrirational parametric Yang-Baxter maps with $3\times 3$ Lax matrices by introducing additional essential parameters. These maps preserve a prescribed Poisson structure which originates from the Sklyanin bracket. We investigate various low-dimensional reductions of this family, as well as degenerate limits with respect to the parameters that were introduced. As a result, we derive several birational Yang-Baxter maps, and we discuss some of their integrability properties. This work is part of a more general classification of Yang-Baxter maps admitting a strong $3\times 3$ Lax matrix with a linear dependence on the spectral parameter.