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arXiv · 1808.05261

The Racah algebra as a commutant and Howe duality

Julien Gaboriaud·Luc Vinet·Stéphane Vinet·Alexei Zhedanov
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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 $\big(\mathfrak{o}(6)$, $\mathfrak{su}(1,1)\big)$.

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Julien Gaboriaud, Luc Vinet, Stéphane Vinet, Alexei Zhedanov. 2018-08-15. The Racah algebra as a commutant and Howe duality. https://doi.org/10.1088/1751-8121/aaee1a

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