The Askey-Wilson algebra and its avatars

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.

arXiv comment34 pages; text updated to the published version, a few typos corrected and references updated

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Cited by in corpus (13)

• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme
• Askey-Wilson braid algebra and centralizer of $U_q({\mathfrak{sl}}_2)$
• The missing label of ${\mathfrak{su}}_3$ and its symmetry
• Computation of entanglement entropy in inhomogeneous free fermions chains by algebraic Bethe ansatz
• Spin models and distance-regular graphs of $q$-Racah type
• Meta Algebras and Biorthogonal Rational Functions: The Hahn Case
• Time and band limiting operator and Bethe ansatz
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Chern-Simons theory, link invariants and the Askey-Wilson algebra
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions