Double Affine Hecke Algebras of Rank 1 and the $Z_3$-Symmetric Askey-Wilson Relations

Abstract

We consider the double affine Hecke algebra $H=H(k_0,k_1,k_0^{\vee },k_1^{\vee };q)$ associated with the root system $(C_1^{\vee },C_1)$. We display three elements $x$, $y$, $z$ in $H$ that satisfy essentially the $Z_3$-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra $\hat{H}$ that is more general than $H$, called the universal double affine Hecke algebra of type $(C_1^{\vee },C_1)$. An advantage of $\hat{H}$ over $H$ is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism $\hat{H}\to H$. We define some elements $x$, $y$, $z$ in $\hat{H}$ that get mapped to their counterparts in $H$ by this homomorphism. We give an action of Artin's braid group $B_3$ on $\hat{H}$ that acts nicely on the elements $x$, $y$, $z$; one generator sends $x\mapsto y\mapsto z\mapsto x$ and another generator interchanges $x$, $y$. Using the $B_3$ action we show that the elements $x$, $y$, $z$ in $\hat{H}$ satisfy three equations that resemble the $Z_3$-symmetric Askey-Wilson relations. Applying the homomorphism $\hat{H}\to H$ we find that the elements $x$, $y$, $z$ in $H$ satisfy similar relations.

References in corpus (3)

• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Raising and Lowering Operators for Askey-Wilson Polynomials

Cited by in corpus (13)

• A "missing" family of classical orthogonal polynomials
• The Universal Askey-Wilson Algebra
• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$
• The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee },C_1)$
• The Askey-Wilson algebra and its avatars
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Center of the universal Askey--Wilson algebra at roots of unity
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• Evaluation modules for the $q$-tetrahedron algebra
• Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle