The Askey-Wilson algebra and its avatars. (English) Zbl 1519.81276
Summary: 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 \((C^\vee_1, C_1)\). 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.
MSC:
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
Askey-Wilson algebra; Kauffman bracket skein algebra; \(U_q(\mathfrak{sl}_2)\) algebra; double affine Hecke algebra; universal \(R\)-matrix; \(W(D_4)\) Weyl group; half Dehn twistReferences:
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