The Racah algebra as a commutant and Howe duality


الملخص بالإنكليزية

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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