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

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 ($K$-list), by considering the symmetries of the $K$-list maps, we obtain compatible refactorization problems with rational Lax matrices for other classes of non-abelian involutive Yang–Baxter maps ($\Lambda$, $H$ and $F$ 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.