On superintegrability

The notion of superintegrability in Hamiltonian mechanics generalizes Liouville integrability in a natural way. Roughly, superintegrable systems have an extra "hidden symmetry". Historically, the first example of such a system is the Kepler system with an extra symmetry generated by what is known now as Lenz vectors. The first part of the talk will be focused on geometric structures related to superintegrability. The second part aims at the corresponding quantum notion. It turns out that known constructions from Lie theory and representation theory typically produce superintegrable systems.