Structures algébriques, systèmes superintégrables et polynômes orthogonaux

This thesis is divided into five parts concerned with the following topics: the physical and algebraic interpretation of families of multivariate orthogonal functions and their applications, the study of superintegrable quantum systems in two and three dimensions involving reflection operators, the characterization of families of orthogonal polynomials of the Bannai-Ito scheme and the study of the algebraic structures associated to them, the investigation of the relationship between the recoupling of irreducible representations of algebras and superalgebras and superintegrable systems, as well as the algebraic interpretation of families of matrix multi-orthogonal polynomials. In the first part, we develop the physical and algebraic interpretation of the Krawtchouk, Meixner and Charlier families of multivariate orthogonal polynomials as matrix elements of unitary representations of the SO(d + 1), SO(d, 1) and E(d) groups on oscillator states. We determine the transition amplitudes between the states of the singular oscillator associated to the Cartesian and polyspherical bases in terms of the multivariate Hahn polynomials. We examine the 9j coefficients of su(1,1) through the generic superintegrable system on the 3-sphere. We characterize the q-Krawtchouk polynomials as matrix elements of "q-rotations" of U_q(sl_2). We show how to design a two-dimensional spin network that allows perfect state transfer using the two-variable Krawtchouk polynomials and we construct a discrete model of the two-dimensional quantum oscillator using the two-variable Meixner polynomials. In the second part, we study superintegrable systems of Dunkl type, which involve reflections. We examine the Dunkl oscillator in two and three dimensions, the singular Dunkl oscillator in the plane and the generic system on the 2-sphere with reflections. We show that each of these systems is superintegrable. We obtain their constants of motion, we find their symmetry algebras as well as their representations, we give their exact solutions and we exhibit their relationship with the orthogonal polynomials of the Bannai-Ito scheme. In the third part, we characterize two families of polynomials belonging to the Bannai-Ito scheme: the complementary Bannai-Ito polynomials and the Chihara polynomials. We also show that the Bannai–Ito polynomials arise as Racah coefficients for the osp(1,2) superalgebra. We determine the symmetry algebra associated with the dual − 1 Hahn polynomials in the context of the Clebsch-Gordan problem for osp(1,2). We introduce a q -generalization of the Bannai-Ito polynomials by examining the Racah problem for the quantum superalgebra osp_q(1,2). Finally, we show that the q-deformed Bannai-Ito algebra serves as a covariance algebra for osp_q(1,2). In the fourth part, we determine the relationship between the recoupling of representations of the su(1,1) and osp(1,2) algebras and second-order superintegrable systems with or without reflections. We also study representations of Racah–Wilson and Bannai-Ito algebras. Moreover, we show that the Racah Wilson algebra serves as a quadratic covariance algebra for sl(2). In the fifth part, we explicitly construct two families of d-orthogonal polynomials based on su(2). We investigate the squeezed/coherent states of the finite oscillator and we characterize a family of matrix multi-orthogonal polynomials.

Keyphrases: Tableau de Bannai–Ito, Orthogonal polynomials, Superintegrable systems, Quadratic algebras, Bannai–Ito scheme, Dunkl operators