On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps

TL;DR

This paper develops rational Lax representations for various classes of quadrirational Yang-Baxter maps, using symmetries and refactorization techniques, expanding the known integrable structures in both abelian and non-abelian settings.

Contribution

It introduces a systematic method to derive Lax matrices for new classes of Yang-Baxter maps via symmetry considerations, extending existing lists.

Findings

Constructed Lax representations for non-abelian involutive Yang-Baxter maps.

Generated Lax matrices for abelian Yang-Baxter maps of the F and H lists.

Provided examples of multi-parametric Yang-Baxter maps outside known classifications.

Abstract

We present rational Lax representations for one-component parametric quadrirational Yang-Baxter maps in both the abelian and non-abelian settings. We show that from the Lax matrices of a general class of non-abelian involutive Yang-Baxter maps ($\mathcal{K}$-list), by considering the symmetries of the $\mathcal{K}$-list maps, we obtain compatible refactorization problems with rational Lax matrices for other classes of non-abelian involutive Yang-Baxter maps ($\Lambda$, $\mathcal{H}$ and $\mathcal{F}$ lists). In the abelian setting, this procedure generates rational Lax representations for the abelian Yang-Baxter maps of the $F$ and $H$ lists. Additionally, we provide examples of non-involutive (abelian and non-abelian) multi-parametric Yang-Baxter maps, along with their Lax representations, which lie outside the preceding lists.