Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(2)

Abstract

The Askey-Wilson algebra $AW(3)$ with three generators is shown to serve as a hidden symmetry algebra underlying the Racah and (new) generalized Clebsch-Gordan problems for the quantum algebra $s{l}_q(2)$. On the base of this hidden symmetry a simple method to calculate corresponding coefficients in terms of the Askey-Wilson polynomials is proposed.

arXiv comment12 pages

Cited by in corpus (21)

• Universal Superpositions of Coherent States and Self-Similar Potentials
• 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 generalized Racah algebra as a commutant
• The Askey-Wilson algebra and its avatars
• 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
• 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
• The Racah algebra as a commutant and Howe duality
• A higher rank extension of the Askey-Wilson Algebra
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager algebra
• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Factorized $A_2$-Leonard pair
• Chern-Simons theory, link invariants and the Askey-Wilson algebra
• Braid group and $q$-Racah polynomials