Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra

Abstract

This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra.

arXiv commentThis is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ In v2 remarks about duality anti-algebra isomorphism and about shift operators added

References in corpus (5)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Double affine Hecke algebras of rank 1 and affine cubic surfaces
• Askey-Wilson polynomials: an affine Hecke algebraic approach
• Raising and Lowering Operators for Askey-Wilson Polynomials

Cited by in corpus (26)

• The Universal Askey-Wilson Algebra
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• An embedding of the universal Askey-Wilson algebra into $U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)$
• The Askey-Wilson algebra and its avatars
• Double Affine Hecke Algebras of Rank 1 and the $Z_3$-Symmetric Askey-Wilson Relations
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• Dualities in the $q$-Askey scheme and degenerate DAHA
• FRT presentation of classical Askey-Wilson algebras
• Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Center of the universal Askey--Wilson algebra at roots of unity
• A classification of sharp tridiagonal pairs
• Totally bipartite tridiagonal pairs
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager algebra
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• Note on character varieties and cluster algebras
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• On the shape of a tridiagonal pair
• An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions