Universal chain structure of quadratic algebras for superintegrable systems with coalgebra symmetry

In this paper we show that the chain structure of (overlapping) quadratic algebras, recently introduced in Liao et al (2018 J. Phys. A: Math. Theor. 51 255201) in the analysis of the nD quantum quasi-generalized Kepler- Coulomb system, naturally arises for nD Hamiltonian systems endowed with an sl(2, R) coalgebra symmetry. As a consequence of this hidden symmetry, in fact, such systems are automatically endowed with 2n - 3 (secondorder) functionally/algebraically independent classical/quantuin conserved quantities arising as the image, through a given symplectic/differential representation, of the so-called left and right Casimirs of the coalgebra. These integrals, which are said to be universal being in common to the entire coalgebraic family of Hamiltonians, are shown to be the building blocks of the overlapping quadratic algebras mentioned above. For this reason a subalgebra of these quadratic structures turns out to be, as a matter of fact, universal in the sense that it is shared by any Hamiltonian belonging to this class. As a new specific result arising from this observation, we present the chain structure of quadratic algebras for the nD quasi-generalized Kepler-Coulomb system on the n-sphere S-k(n) and on the hyperbolic n-space H-k(n) Both the classical and the quantum frameworks will be considered.