Poisson Yang-Baxter maps with binomial Lax matrices

TL;DR

This paper constructs multidimensional parametric Yang-Baxter maps using binomial Lax matrices, demonstrating their symplectic and integrable properties, and introduces a family of quadrirational symplectic YB maps with specific matrix structures.

Contribution

It presents a novel construction of Yang-Baxter maps with binomial Lax matrices and explores their symplectic and integrable features, including a new family of quadrirational maps.

Findings

Maps are symplectic with respect to reduced symplectic structures.

Construction of multidimensional parametric Yang-Baxter maps.

Introduction of quadrirational symplectic YB maps with 3x3 Lax matrices.

Abstract

A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on $\mathbb{C}^4 \times \mathbb{C}^4$ with $3\times 3$ Lax matrices is also presented.