An embedding of the universal Askey-Wilson algebra into $U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)$

Abstract

The Askey--Wilson algebras were used to interpret the algebraic structure hidden in the Racah--Wigner coefficients of the quantum algebra $U_q({\mathfrak{sl}}_2)$. In this paper, we display an injection of a universal analog ${△}_q$ of Askey--Wilson algebras into $U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)$ behind the application. Moreover we formulate the decomposition rules for $3$-fold tensor products of irreducible Verma $U_q({\mathfrak{sl}}_2)$-modules and of finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules into the direct sums of finite-dimensional irreducible ${△}_q$-modules.

References in corpus (4)

• 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
• From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation

Cited by in corpus (17)

• The Askey-Wilson algebra and its avatars
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• 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
• 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 $U_q({\mathfrak{sl}}_2)$
• The $q$-Higgs and Askey-Wilson algebras
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• 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
• An Askey-Wilson Algebra of Rank 2
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$
• Spin models and distance-regular graphs of $q$-Racah type
• Braid group and $q$-Racah polynomials
• Chern-Simons theory, link invariants and the Askey-Wilson algebra