Structure relations for the symmetry algebras of quantum superintegrable systems

A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with potential H = Delta(n) + V that admits 2n - 1 algebraically independent partial differential operators commuting with the Hamiltonian, the maximum number possible. Here, n >= 2. The system is of order l if the maximum order of the symmetry operators other than the Hamiltonian is l. Typically, the algebra generated by the symmetry operators has been shown to close. There is an analogous definition for classical superintegrable systems with the operator commutator replaced by the Poisson bracket. Superintegrability captures what it means for a Hamiltonian system to be explicitly algebraically and analytically solvable, not just solvable numerically. Until recently there were very few examples of superintegrable systems of order l with l > 3 and and virtually no structure results. The situation has changed dramatically in the last two years with the discovery of families of systems depending on a rational parameter k - p/q that are superintegrable for all k and of arbitrarily high order, such as l = p + q + 1. We review a method, based on recurrence formulas for special functions, that proves superintegrability of these higher order quantum systems, and allows us to determine the structure of the symmetry algebra. Just a few months ago, these constructions seemed out of reach.