The Racah algebra as a commutant and Howe duality

Abstract

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2)\oplus \mathfrak{o}(2)\oplus \mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(6)$ in oscillator representations of the universal algebra of the latter. How this observation relates to the $\mathfrak{su}(1,1)$ Racah problem and the superintegrable model on the $2$-sphere is discussed on the basis of the Howe duality associated to the pair $(\mathfrak{o}(6)$, $\mathfrak{su}(1,1))$.

arXiv comment9 pages

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Cited by in corpus (8)

• The Higgs and Hahn algebras from a Howe duality perspective
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• 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
• Howe duality and algebras of the Askey-Wilson type: an overview
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients