The AskeyWilson algebra and its avatars
Abstract
The original AskeyWilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name AskeyWilson 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 fourpunctured 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 Rmatrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank AskeyWilson algebras are briefly discussed and summarized in a diagrammatic fashion.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 February 2021
 DOI:
 10.1088/17518121/abd783
 arXiv:
 arXiv:2009.14815
 Bibcode:
 2021JPhA...54f3001C
 Keywords:

 AskeyWilson algebra;
 Kauffman bracket skein algebra;
 ${U}_{q}\left({\mathfrak{s}\mathfrak{l}}_{2}\right)$Uq(sl2) algebra;
 double affine Hecke algebra;
 universal Rmatrix;
 W(D4) Weyl group;
 half Dehn twist;
 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Rings and Algebras
 EPrint:
 34 pages