The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Abstract

Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

arXiv commentThis is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ In v4 a slight error in formula (2.8) for Q_0 is corrected

References in corpus (5)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• The structure relation for Askey-Wilson polynomials
• Askey-Wilson polynomials: an affine Hecke algebraic approach
• Raising and Lowering Operators for Askey-Wilson Polynomials
• Koornwinder polynomials and affine Hecke algebras

Cited by in corpus (40)

• The Universal Askey-Wilson Algebra
• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee },C_1)$
• 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
• Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$
• FRT presentation of classical Askey-Wilson algebras
• Center of the universal Askey--Wilson algebra at roots of unity
• Totally bipartite tridiagonal pairs
• The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials
• Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
• A classification of sharp tridiagonal pairs
• Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager algebra
• Tridiagonal Symmetries of Models of Nonequilibrium Physics
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• Higher Rank Askey-Wilson Algebras as Skein Algebras
• Generalized double affine Hecke algebras, their representations, and higher Teichmüller theory
• On the shape of a tridiagonal pair
• Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms
• Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• Non-Symmetric Askey--Wilson Shift Operators
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle
• Symmetric abstract hypergeometric polynomials
• Jackson's $(-1)$-Bessel functions with the Askey-Wilson algebra setting
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions