The Askey-Wilson algebra and its avatars

TL;DR

This paper reviews various forms of the Askey-Wilson algebra, focusing on two key avatars related to quantum groups and skein algebras, and explores their symmetries, representations, and connections to braid group actions.

Contribution

It clarifies the relationships between different Askey-Wilson algebra variants, introduces two main avatars with their algebraic and geometric properties, and discusses higher rank generalizations.

Findings

The first avatar is invariant under the Weyl group of type D4.

The second avatar is isomorphic to the Kauffman bracket skein algebra of the four-punctured sphere.

Connections between the second avatar, the universal double affine Hecke algebra, and the Racah problem are established.

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.