Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps

TL;DR

This paper introduces two non-Abelian hierarchies of compatible maps and their Lax pairs, linking them to non-Abelian Yang-Baxter maps and integrable difference systems on a lattice, including new hierarchies like the lattice-NQC Gel'fand-Dikii.

Contribution

It presents novel non-Abelian hierarchies of compatible maps, their Lax pair formulations, and explicit non-Abelian Yang-Baxter maps, expanding the understanding of integrable lattice systems.

Findings

Introduced two non-Abelian hierarchies of compatible maps.

Established their connection to non-Abelian Yang-Baxter maps.

Derived explicit forms of non-Abelian lattice Gel'fand-Dikii hierarchies.

Abstract

We present two non-equivalent families of hierarchies of non-Abelian compatible maps and we provide their Lax pair formulation. These maps are associated with families of hierarchies of non-Abelian Yang-Baxter maps, which we provide explicitly. In addition, these hierarchies correspond to integrable difference systems with variables defined on edges of an elementary cell of the $\mathbb{Z}^2$ graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. In that respect we obtain the non-Abelian lattice-modified Gel'fand-Dikii hierarchy, together with the explicit form of a non-Abelian hierarchy that we refer to as the lattice-NQC (or lattice-$(Q3)_0$) Gel'fand-Dikii hierarchy.