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

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.

29 pages, 2 figures. v2:Typos corrected, one figure added

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Cited by in corpus (2)

• Matrix factorizations and pentagon maps
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps