The Universal Askey-Wilson Algebra

Abstract

In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $\Delta$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\mathrm{PSL}}_2(\mathbb{Z})$ on $\Delta$ as a group of automorphisms. We give a linear basis for $\Delta$. We describe the center of $\Delta$ and the 2-sided ideal $\Delta [\Delta ,\Delta ]\Delta$. We discuss how $\Delta$ is related to the $q$-Onsager algebra.

24 pages

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Cited by in corpus (44)

• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$
• The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee },C_1)$
• 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
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• The alternating central extension of the $q$-Onsager algebra
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• 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
• Lie polynomials in $q$-deformed Heisenberg algebras
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Generalized q-Onsager Algebras and Dynamical K-matrices
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• Center of the universal Askey--Wilson algebra at roots of unity
• A conjecture concerning the $q$-Onsager algebra
• Totally bipartite tridiagonal pairs
• The $q$-Higgs and Askey-Wilson algebras
• 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
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• A Lie algebra related to the universal Askey-Wilson algebra
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• A Casimir element inexpressible as a Lie polynomial
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• Spin models and distance-regular graphs of $q$-Racah type
• Higher Rank Askey-Wilson Algebras as Skein Algebras
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms
• Symmetric abstract hypergeometric polynomials
• Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra
• Centrification of Algebras and Hopf Algebras
• Double Lowering Operators on Polynomial
• Representations of Askey--Wilson algebra
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform