The Askey-Wilson algebra and its avatars

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 D₄ 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 $\left({C}_{1}^{\vee },{C}_{1}\right)$ . This second algebra emerges from the Racah problem of ${U}_{q}\left({\mathfrak{s}\mathfrak{l}}_{2}\right)$ and is related via an injective homomorphism to the centralizer of ${U}_{q}\left({\mathfrak{s}\mathfrak{l}}_{2}\right)$ 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.