The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms
We propose a definition by generators and relations of the rank − 2 Askey-Wilson algebra () for any integer , generalising the known presentation for the usual case =3. The generators are indexed by connected subsets of {1, …, }, and the simple and rather small set of defining relations is directl...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212007 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms. Nicolas Crampé, Luc Frappat, Loïc Poulain d'Andecy and Eric Ragoucy. SIGMA 19 (2023), 077, 36 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862737579257888768 |
|---|---|
| author | Crampé, Nicolas Frappat, Luc Poulain d'Andecy, Loïc Ragoucy, Eric |
| author_facet | Crampé, Nicolas Frappat, Luc Poulain d'Andecy, Loïc Ragoucy, Eric |
| citation_txt | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms. Nicolas Crampé, Luc Frappat, Loïc Poulain d'Andecy and Eric Ragoucy. SIGMA 19 (2023), 077, 36 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We propose a definition by generators and relations of the rank − 2 Askey-Wilson algebra () for any integer , generalising the known presentation for the usual case =3. The generators are indexed by connected subsets of {1, …, }, and the simple and rather small set of defining relations is directly inspired by the known case of = 3. Our first main result is to prove the existence of automorphisms of () satisfying the relations of the braid group on + 1 strands. We also show the existence of coproduct maps relating the algebras for different values of . An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the -fold tensor product of the quantum group Uq(₂) or, equivalently, onto the Kauffman bracket skein algebra of the ( + 1)-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to 0 in the realisation in the -fold tensor product of Uq(₂), thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.
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| first_indexed | 2026-04-17T16:56:27Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212007 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T16:56:27Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Crampé, Nicolas Frappat, Luc Poulain d'Andecy, Loïc Ragoucy, Eric 2026-01-22T09:15:59Z 2023 The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms. Nicolas Crampé, Luc Frappat, Loïc Poulain d'Andecy and Eric Ragoucy. SIGMA 19 (2023), 077, 36 pages 1815-0659 2020 Mathematics Subject Classification: 16T10; 33D45; 81R12 arXiv:2303.17677 https://nasplib.isofts.kiev.ua/handle/123456789/212007 https://doi.org/10.3842/SIGMA.2023.077 We propose a definition by generators and relations of the rank − 2 Askey-Wilson algebra () for any integer , generalising the known presentation for the usual case =3. The generators are indexed by connected subsets of {1, …, }, and the simple and rather small set of defining relations is directly inspired by the known case of = 3. Our first main result is to prove the existence of automorphisms of () satisfying the relations of the braid group on + 1 strands. We also show the existence of coproduct maps relating the algebras for different values of . An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the -fold tensor product of the quantum group Uq(₂) or, equivalently, onto the Kauffman bracket skein algebra of the ( + 1)-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to 0 in the realisation in the -fold tensor product of Uq(₂), thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements. The authors thank A. Lacabanne for fruitful discussions. N. Crampe and L. Poulain d’Andecy thank LAPTh for its hospitality and are supported by the international research project AAPT of the CNRS and the ANR Project AHA ANR-18-CE40-000. This work is partially supported by Université Savoie Mont Blanc and Conseil Savoie Mont Blanc grant APOINT. We also wish to thank the anonymous referees for their valuable comments during the course of revision. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms Article published earlier |
| spellingShingle | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms Crampé, Nicolas Frappat, Luc Poulain d'Andecy, Loïc Ragoucy, Eric |
| title | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms |
| title_full | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms |
| title_fullStr | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms |
| title_full_unstemmed | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms |
| title_short | The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms |
| title_sort | higher-rank askey-wilson algebra and its braid group automorphisms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212007 |
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