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

Pavlos Kassotakis, Theodoros E. Kouloukas, Maciej Nieszporski

TL;DR

The work develops rational Lax representations for parametric quadrirational Yang–Baxter maps in both abelian and non‑abelian settings by leveraging refactorization problems tied to Lax matrices and exploiting the symmetry structure of the non‑abelian $\mathcal{K}$ list to generate Lax data for the $\Lambda$, $\mathcal{H}$, and $\mathcal{F}$ lists. In the abelian limit, the approach yields rational Lax matrices for the $H$‑ and $F$‑lists, with explicit constructions and a discussion of which cases preserve rationality (notably excluding $F_{IV}$ due to symmetry obstructions). The paper also provides explicit non‑abelian six‑parameter extensions with associated Lax representations, and outlines a path toward a more complete classification of quadrirational YB maps in the abelian setting. Overall, the results deepen the link between symmetry, refactorization, and Lax representations in the theory of Yang–Baxter maps, with implications for discrete integrable systems and their algebraic structures.

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 ($Λ$, $\mathcal{H}$ 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.

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

TL;DR

The work develops rational Lax representations for parametric quadrirational Yang–Baxter maps in both abelian and non‑abelian settings by leveraging refactorization problems tied to Lax matrices and exploiting the symmetry structure of the non‑abelian list to generate Lax data for the , , and lists. In the abelian limit, the approach yields rational Lax matrices for the ‑ and ‑lists, with explicit constructions and a discussion of which cases preserve rationality (notably excluding due to symmetry obstructions). The paper also provides explicit non‑abelian six‑parameter extensions with associated Lax representations, and outlines a path toward a more complete classification of quadrirational YB maps in the abelian setting. Overall, the results deepen the link between symmetry, refactorization, and Lax representations in the theory of Yang–Baxter maps, with implications for discrete integrable systems and their algebraic structures.

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 (-list), by considering the symmetries of the -list maps, we obtain compatible refactorization problems with rational Lax matrices for other classes of non-abelian involutive Yang-Baxter maps (, and lists). In the abelian setting, this procedure generates rational Lax representations for the abelian Yang-Baxter maps of the and 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.
Paper Structure (7 sections, 5 theorems, 37 equations, 1 figure, 2 tables)

This paper contains 7 sections, 5 theorems, 37 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Yang--Baxter maps and Lax matrices
  3. The H and the F list of quadrirational YB maps and their symmetries
  4. Lax matrices of YB maps which are related through symmetry
  5. Rational Lax matrices for the $\mathcal{K},\Lambda, \mathcal{H}$ and $\mathcal{F}$ lists of non-abelian quadrirational YB maps
  6. Discussion
  7. The $F$ and the $H-$list of quadrirational Yang-Baxter maps

Key Result

Proposition 1.1

We consider a parametric map $R^{p,q}: \mathbb{X} \times \mathbb{X} \rightarrow \mathbb{X} \times \mathbb{X}$ with an effective$L\in GL_N$ acts identically on $\mathbb{X}$, iff $L=Id$ action $GL_N \times \mathbb{X}\rightarrow \mathbb{X}$ and a matrix valued function $L: \mathbb{X}\times \mathbb{I} \times \mathbb{I} \rightarrow GL_N$ such that where $L[x]$ denotes the action of $L\in GL_N$ on $x

Figures (1)

  • Figure 1: The $\mathcal{F},$$\mathcal{H},$$\mathcal{K}$ and $\Lambda$ lists of quadrirational Yang-Baxter maps in the non-abelian and in the abelian setting. The generic members of these lists are related by the morphisms $\Phi: R\rightarrow (\phi^{-1}\times id)R(id\times \phi)$ and $\Psi: R\rightarrow (\psi^{-1}\times id )R(id\times \psi),$ where $\phi,\psi,$ symmetries.

Theorems & Definitions (10)

  • Proposition 1.1: ves4
  • Proposition 1.2: PSTV
  • Definition 1.3: PSTV
  • Definition 1.4: ABS2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4