A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients

Abstract

Two descriptions of the dual $-1$ Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra $\mathfrak{o}(4)$ and the Lie superalgebra $\mathfrak{osp}(1|2)$.

arXiv comment12 pages

References in corpus (25)

• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• Deformed Dolan-Grady relations in quantum integrable models
• Dual pairing of symmetry groups and dynamical groups in physics
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• The Bannai-Ito algebra and some applications
• A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• From $s{l}_q(2)$ to a Parabosonic Hopf Algebra
• The generalized Racah algebra as a commutant
• Dunkl shift operators and Bannai-Ito polynomials
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$
• The Higgs and Hahn algebras from a Howe duality perspective
• The Racah algebra as a commutant and Howe duality
• The ${\mathbb{Z}}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra
• A higher rank extension of the Askey-Wilson Algebra
• The $q$-Higgs and Askey-Wilson algebras
• Truncation of the reflection algebra and the Hahn algebra
• Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Generating functions for the osp(1|2) Clebsch-Gordan coefficients
• Howe duality and algebras of the Askey-Wilson type: an overview
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics