The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras

Abstract

The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1){)}^{\otimes 3}$. We analyze the commutant of ${\mathfrak{o}}_{q^{1/2}}(2)\oplus {\mathfrak{o}}_{q^{1/2}}(2)\oplus {\mathfrak{o}}_{q^{1/2}}(2)$ in $q$-oscillator representations of ${\mathfrak{o}}_{q^{1/2}}(6)$ and show that it also realizes $AW(3)$. These two pictures of $AW(3)$ are shown to be dual in the sense of Howe; this is made clear by highlighting the role of the intermediate Casimir elements of each members of the dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(6))$. We also generalize these results. A higher rank extension of the Askey-Wilson algebra denoted $AW(n)$ can be defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1){)}^{\otimes n}$ and a dual description of $AW(n)$ as the commutant of ${\mathfrak{o}}_{q^{1/2}}(2{)}^{\oplus n}$ in $q$-oscillator representations of ${\mathfrak{o}}_{q^{1/2}}(2n)$ is offered by calling upon the dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$.

arXiv comment14 pages

References in corpus (23)

• An integrable structure related with tridiagonal algebras
• Deformed Dolan-Grady relations in quantum integrable models
• A new (in)finite dimensional algebra for quantum integrable models
• Dual pairing of symmetry groups and dynamical groups in physics
• Coideal Subalgebras and Quantum Symmetric Pairs
• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• On Dunkl angular momenta algebra
• Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(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 generalized Racah algebra as a commutant
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• The generic quantum superintegrable system on the sphere and Racah operators
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• The Racah algebra as a commutant and Howe duality
• A higher rank extension of the Askey-Wilson Algebra
• On the Casimir Elements of q-Algebras U'_q(so_n) and Their Eigenvalues in Representations
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• Tridiagonal Symmetries of Models of Nonequilibrium Physics

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• The Askey-Wilson algebra and its avatars
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• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Howe duality and algebras of the Askey-Wilson type: an overview
• 2-Homogeneous bipartite distance-regular graphs and the quantum group $U_q^{\prime }({\mathfrak{so}}_6)$
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients