Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Author: Kassotakis, P.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210174
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps / P. Kassotakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 73 назв. — англ.

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps, and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables on the invariants of this family of maps, the HI, HII, and HAIII Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the HI, HII, and HAIII Yang-Baxter maps, whereas by the second method, we obtain non-periodic entwining maps associated with the whole F and H-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the H-list of Yang-Baxter maps can be considered as the (k−1)-iteration of some maps of simpler form. We refer to these maps as extended transfer maps, and in turn, they lead to k-point alternating recurrences, which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.