The generic quantum superintegrable system on the sphere and Racah operators

Abstract

We consider the generic quantum superintegrable system on the d-sphere with potential V (y) = d+1 k=1 b k y 2 k , where b k are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in d variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in [9]. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik's multivariable Racah polynomials.

A Discrete Realization of the Higher Rank Racah Algebra

“…The explicit form of the Racah polynomials can be determined by the action of the generators of R (3). For that aim we introduce the polynomial (11) κ(x, β)…”

“…It thus becomes very natural to extend the Racah algebra to higher rank. This was initially done in the superintegrable context [11] using the n-dimensional generalization of (3), and the irreducibility of the representation spaces that arise in this model has been proven in [12]. In the context of abstract tensor products, the construction was given in [2] and illustrated with a new model using the (Z 2 ) n Laplace-Dunkl operator [3,4].…”

Darboux transformations from the Appell-Lauricella operator

“…The system with Hamiltonian H in (3.7) has been extensively studied in the literature as an important example of a second-order superintegrable system, possessing the maximal possible number of algebraically independent second-order integrals of motion. It is usually referred to as the generic quantum superintegrable system on the sphere, and has attracted a lot of attention recently in connection to multivariate extensions of the Askey scheme of hypergeometric orthogonal polynomials and their bispectral properties, the Racah problem for su (1, 1), representations of the Kohno-Drinfeld algebra, the Laplace-Dunkl operator associated with Z d+1 2 root system; see for instance [8,16,19] and the references therein. The space V d n introduced in Remark 3.3 appears naturally in the analysis as an irreducible module over the associative algebra generated by the integrals of motion, see [17].…”

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

“…Note that we have replaced 1 ⊗ ∆(Γ q ) by the rank 1 operator Γ q {2,3} . Observe that the sequences of applied extension morphisms in (28) and (29) are identical. The same sequence also arises when constructing Γ q…”