Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
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 Yan...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210174 |
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| 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862553528880332800 |
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| author | Kassotakis, P. |
| author_facet | Kassotakis, P. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T21:24:38Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210174 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:38Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kassotakis, P. 2025-12-03T14:24:39Z 2019 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. 1815-0659 2010 Mathematics Subject Classification: 14E07; 14H70; 37K10 arXiv: 1901.01609 https://nasplib.isofts.kiev.ua/handle/123456789/210174 https://doi.org/10.3842/SIGMA.2019.048 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. P.K. is grateful to Aristophanis Dimakis, Vassilios Papageorgiou, and Anastasios Tongas, the organizers of the 4th Workshop on Mathematical Physics-Integrable Systems (November 30 -December 1, 2018, Department of Mathematics, University of Patras, Patras, Greece), where this work was finalized. Also, P.K. is grateful to James Atkinson, Allan Fordy, Nalini Joshi, Frank Nijhoff, and Pol Vanhaecke for very fruitful discussions on the subject, as well as to Maciej Nieszporski for the endless discussions towards the answer to the great question of integrable systems, Yang-Baxter maps, and everything. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps Article published earlier |
| spellingShingle | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps Kassotakis, P. |
| title | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps |
| title_full | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps |
| title_fullStr | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps |
| title_full_unstemmed | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps |
| title_short | Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps |
| title_sort | invariants in separated variables: yang-baxter, entwining and transfer maps |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210174 |
| work_keys_str_mv | AT kassotakisp invariantsinseparatedvariablesyangbaxterentwiningandtransfermaps |