The Askey-Wilson algebra and its avatars
| Autor: | Crampé, Nicolas, Frappat, Luc, Gaboriaud, Julien, d'Andecy, Loïc Poulain, Ragoucy, Eric, Vinet, Luc |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | J. Phys. A 54 (2021), no. 6, Paper No. 063001, 32 pp |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8121/abd783 |
| Popis: | The original Askey-Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name 'Askey-Wilson algebra' is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type $D_4$ and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra $(C_1^{\vee},C_1)$. This second algebra emerges from the Racah problem of $U_q(\mathfrak{sl}_2)$ and is related via an injective homomorphism to the centralizer of $U_q(\mathfrak{sl}_2)$ in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by $R$-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey-Wilson algebras are briefly discussed and summarized in a diagrammatic fashion. Comment: 34 pages; text updated to the published version, a few typos corrected and references updated |
| Databáze: | arXiv |
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