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 , C 1 ). This second algebra emerges from the Racah problem of U q (sl 2 ) and is related via an injective homomorphism to the centralizer of U q (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.Keywords: Askey-Wilson algebra, Kauffman bracket skein algebra, U q (sl 2 ) algebra, double affine Hecke algebra, centralizer, universal R-matrix, W (D 4 ) Weyl group, half Dehn twist.

Chern-Simons theory, link invariants and the Askey-Wilson algebra

“…The strategy for establishing the role of the Askey-Wilson algebra in the CS theory will rely on connecting the expectation values of the Wilson loops in the fundamental representation of su 2 with the Kauffman bracket polynomial. It is relevant to point out a parallel with the relation between the AW algebra and the Kauffman skein algebra that was obtained in [5,6,8,29].…”

“…More specifically, one can use the property (iv) of the bracket polynomial to simplify the crossings of the diagrams. Such a computation can be found in [8], but we reproduce it here:…”

“…In particular, when considering three spins 1/2 or three spins 1, respectively, the TL and BMW algebras are recovered as quotients of the AW algebra. In the context of knot theory, the AW algebra is connected to the Kauffman skein algebra of (framed and unoriented) links in some punctured surfaces, see [5,6,8,29].…”

“…The strategy we follow in this paper is to associate to the generators of the AW algebra some tangle diagrams composed of three straight strands with a subset of them being enclosed by a loop associated to the spin 1/2 representation. These are inspired by the diagrams of the Kauffman skein algebra in [8], where punctures on a plane are enclosed by loops. Then, we show that the values of the link invariants of these diagrams satisfy the defining relations of the AW algebra both (and independently) in the CS theory and the RT construction.…”

Time and band limiting operator and Bethe ansatz

“…These relations define an algebra which is a particular case of the Askey-Wilson algebra appearing in the context of the eponymous polynomials [43] or in the Racah problem for U q (sl 2 ) [27] (see e.g. [18] for a review). In the context of association schemes, this algebra corresponds to the irreducible decomposition of the Terwilliger algebra associated to the 2n-gon.…”