  Two-dimensional quadratic algebras are generalizations of Lie algebras that
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. Distinct superintegrable
systems and their quadratic algebras can be related by geometric contractions,
induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric
contractions have important physical and geometric meanings, such as obtaining
classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and
nonrelativistic phenomena from special relativistic as $c\to \infty$, and the
derivation of the Askey scheme for obtaining all hypergeometric orthogonal
polynomials as limits of Racah/Wilson polynomials. In this paper we show how to
simplify the structure relations for abstract nondegenerate and degenerate
quadratic algebras and their contractions. In earlier papers we have classified
contractions of 2nd order superintegrable systems on constant curvature spaces
and have shown that all results are derivable from free quadratic algebras
contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in
flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The
quadratic algebra contractions are induced by generalizations of
In\"on\"u-Wigner contractions of these Lie algebras. As a special case we
obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we
complete this theoretical development for 2D superintegrable systems by showing
that the Darboux superintegrable systems are also characterized by free
quadratic algebras contained in the symmetry algebras of these spaces and that
their contractions are also induced by In\"on\"u-Wigner contractions. We
present tables of the contraction results.
