The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$

Abstract

Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension $\Delta$ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between $\Delta$ and the quantum algebra $U_q({\mathfrak{sl}}_2)$. Our main result is an algebra injection from $\Delta$ into a relative of $U_q({\mathfrak{sl}}_2)$; the relative is obtained from $U_q({\mathfrak{sl}}_2)$ by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of $U_q({\mathfrak{sl}}_2)$

misprint in Lemma 7.1 is corrected

References in corpus (3)

• The Universal Askey-Wilson Algebra
• 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 the $Z_3$-Symmetric Askey-Wilson Relations

Cited by in corpus (22)

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• Bispectrality of the Complementary Bannai-Ito Polynomials
• The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee },C_1)$
• The Askey-Wilson algebra and its avatars
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
• Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type
• Center of the universal Askey--Wilson algebra at roots of unity
• Finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules from the equitable point of view
• An Askey-Wilson Algebra of Rank 2
• Billiard Arrays and finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules
• Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$
• Some $q$-exponential formulas for finite-dimensional ${\square }_q$-modules
• Evaluation modules for the $q$-tetrahedron algebra
• The algebra $U_q({\mathfrak{sl}}_2)$ in disguise
• The positive even subalgebra of $U_q({\mathfrak{sl}}_2)$ and its finite-dimensional irreducible modules
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• The Lusztig automorphism of $U_q({\mathfrak{sl}}_2)$ from the equitable point of view
• Freidel-Maillet type presentations of $U_q(s{l}_2)$
• Double Lowering Operators on Polynomial
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra