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arXiv · 2009.14815

The Askey-Wilson algebra and its avatars

Nicolas Crampé·Luc Frappat·Julien Gaboriaud·Loïc Poulain d'Andecy·Eric Ragoucy·Luc Vinet
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Abstract

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.

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Nicolas Crampé, Luc Frappat, Julien Gaboriaud, Loïc Poulain d'Andecy, Eric Ragoucy, Luc Vinet. 2022-02-22. The Askey-Wilson algebra and its avatars. https://doi.org/10.1088/1751-8121%2Fabd783

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