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

Symmetry Integrability and Geometry-Methods and Applications

article

Pavlos Kassotakis¹ 

[1] Department of Mathematics and Statistics, University of Cyprus

关键词: discrete integrable systems;    Yang–Baxter maps;    entwining maps;    transfer maps;   

DOI  :  10.3842/SIGMA.2019.048

来源: National Academy of Science of Ukraine

【 摘 要 】

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 to the invariants of this family of maps, the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ 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 $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ 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.

【 授权许可】

Unknown