Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices

TL;DR

This paper constructs and classifies a family of Poisson Yang-Baxter maps using 2x2 first-degree polynomial Lax matrices, revealing their integrability and connection to known systems.

Contribution

It introduces a new class of quadrirational Yang-Baxter maps derived from refactorization of matrix polynomials and classifies them via Jordan normal form.

Findings

Maps are Poisson with respect to the Sklyanin bracket.

Explicit formulas for quadrirational maps are provided.

Connections to integrable systems on quad graphs are established.

Abstract

A family of nonparametric Yang Baxter (YB) maps is constructed by refactorization of the product of two 2 by 2 matrix polynomials of first degree. These maps are Poisson with respect to the Sklyanin bracket. For each Casimir function a parametric Poisson YB map is generated by reduction on the corresponding level set. By considering a complete set of Casimir functions symplectic multiparametric YB maps are derived. These maps are quadrirational with explicit formulae in terms of matrix operations. Their Lax matrices are, by construction, 2 by 2 first degree polynomial in the spectral parameter and are classified by Jordan normal form of the leading term. Nonquadrirational parametric YB maps constructed as limits of the quadrirational ones are connected to known integrable systems on quad graphs.