  Quadratic algebras are generalizations of Lie algebras; they include the
symmetry algebras of 2nd order superintegrable systems in 2 dimensions as
special cases. The superintegrable systems are exactly solvable physical
systems in classical and quantum mechanics. For constant curvature spaces we
show that the free quadratic algebras generated by the 1st and 2nd order
elements in the enveloping algebras of their Euclidean and orthogonal symmetry
algebras correspond one-to-one with the possible superintegrable systems with
potential defined on these spaces. We describe a contraction theory for
quadratic algebras and show that for constant curvature superintegrable
systems, ordinary Lie algebra contractions induce contractions of the quadratic
algebras of the superintegrable systems that correspond to geometrical
pointwise limits of the physical systems. One consequence is that by
contracting function space realizations of representations of the generic
superintegrable quantum system on the 2-sphere (which give the structure
equations for Racah/Wilson polynomials) to the other superintegrable systems
one obtains the full Askey scheme of orthogonal hypergeometric polynomials.
