Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$

Abstract

A description of the embedding of the universal Askey--Wilson algebra, AW(3), in $U_q(s{l}_2{)}^{\otimes 3}$ is given in terms of the universal R-matrix of $U_q(s{l}_2)$. The generators of the centralizer of $U_q(s{l}_2)$ in its three-fold product are naturally expressed through conjugations of Casimir elements with R. They are identified as the images of the generators of AW(3) under the embedding map by showing that they obey the AW(3) relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.

arXiv commentProposition 4.1 improved and references added

References in corpus (12)

• Superintegrability in two dimensions and the Racah-Wilson algebra
• The Universal Askey-Wilson Algebra
• 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
• Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(2)
• 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 higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• The ${\mathbb{Z}}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra
• A higher rank extension of the Askey-Wilson Algebra
• Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra
• Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)

Cited by in corpus (9)

• The Askey-Wilson algebra and its avatars
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• Askey-Wilson braid algebra and centralizer of $U_q({\mathfrak{sl}}_2)$
• An Askey-Wilson Algebra of Rank 2
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Chern-Simons theory, link invariants and the Askey-Wilson algebra
• Braid group and $q$-Racah polynomials