Entwining Yang-Baxter maps and integrable lattices

TL;DR

This paper introduces Yang-Baxter map systems with zero curvature representations, linking them to integrable lattice structures and revealing their spectral-preserving properties through multidimensional mappings.

Contribution

It presents a novel framework connecting Yang-Baxter maps with integrable lattices via Lax triples and zero curvature representations, expanding understanding of their compatibility and spectral properties.

Findings

Yang-Baxter maps admit zero curvature representations with spectral parameters.

Compatibility conditions lead to a 3D integrability property.

Periodic initial value problems produce spectrum-preserving multidimensional mappings.

Abstract

Yang-Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 x 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L1 = L2 = L3 this property yields the se--theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic 'staircase' initial value problems on quadrilateral lattices, these maps give rise to multidimensional integrable mappings which preserve the spectrum of the corresponding monodromy matrix.