Entwining Yang-Baxter maps and integrable lattices
Volume 93 / 2011
Abstract
Yang–Baxter (YB) map systems (or set-theoretic analogs of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples $L_1,\ L_2,\ L_3$ derived from symplectic leaves of $2 \times 2$ 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 $L_1=L_2=L_3$ this property yields the set-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.
