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 $\big(\mathfrak{o}(6)$, $\mathfrak{su}(1,1)\big)$.