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 ( s l 2 ) and is related via an injective homomorphism to the centralizer of U q ( s l 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.
The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the 2-sphere. It is further identified as the commutant of the o(2) ⊕ o(2) ⊕ o(2) subalgebra of o(6) in oscillator representations of the universal algebra of the latter. How this observation relates to the su(1,1) Racah problem and the superintegrable model on the 2-sphere is discussed on the basis of the Howe duality associated to the pair o(6), su(1,1) .
The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the symmetry algebra of the harmonic oscillator on the 2-sphere. These two algebras are recognized as the commutant of a o(2) ⊕ o(2) subalgebra of o(4) in the oscillator representation of the universal algebra U (u(4)). This connection is further related to the embedding of the (discrete) Hahn algebra in U (su(1,1)) ⊗ U (su(1,1)) in light of the dual action of the pair o(4),su(1,1) on the state vectors of four harmonic oscillators. The two-dimensional singular oscillator is naturally seen by dimensional reduction to have the Higgs algebra as its symmetry algebra.
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