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 -sphere. It is further identified as the commutant of the subalgebra of in oscillator representations of the universal algebra of the latter. How this observation relates to the Racah problem and the superintegrable model on the -sphere is discussed on the basis of the Howe duality associated to the pair .