Generalized Askey-Wilson and q-Onsager algebras: a quantum algebraic approach to multivariate orthogonal polynomials

The Askey-Wilson algebra was introduced as the algebraic structure behind the Askey-Wilson orthogonal polynomials. It is closely related to the q-Onsager algebra, which originates from statistical mechanics. Both algebras appear in the context of superintegrable quantum systems. Such systems are governed by a Hamiltonian which possesses a sufficient number of symmetry operators. These symmetries are invaluable tools to solve the equations of motion determined by the model. The Askey-Wilson algebra has arisen as the symmetry algebra of a quantum system of this kind, where the Hamiltonian is built from so-called q-Dunkl operators. It is also interesting from an algebraic point of view, notably in connection with quantum groups, Hecke algebras and Leonard pairs. In this project, we will study several generalizations of these algebras. These are motivated on the one hand by the extension of the related quantum systems to multiple particles and higher dimensions. On the other hand, these generalizations arise naturally in an algebraic setting, namely from the theory of quantum symmetric pairs. Moreover, these novel algebraic structures allow to extend known connections with orthogonal polynomials to multiple variables. Two multivariate extensions are known for the q-Askey scheme of orthogonal polynomials. We will study the generalized Askey-Wilson and q-Onsager algebras, in the objective of casting both classes of multivariate polynomials in a larger algebraic framework.