Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 043, 33 pages      arXiv:1502.00128      https://doi.org/10.3842/SIGMA.2015.043
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Robin Heinonen a, Ernest G. Kalnins b, Willard Miller Jr. a and Eyal Subag c
a) School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA
b) Department of Mathematics, University of Waikato, Hamilton, New Zealand
c) School of Mathematical Science, Tel Aviv University, Tel Aviv 69978, Israel

Received February 03, 2015, in final form May 30, 2015; Published online June 08, 2015

Abstract
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 Inönü-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 Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. 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 Inönü-Wigner contractions. We present tables of the contraction results.

Key words: contractions; quadratic algebras; superintegrable systems; Darboux spaces; Askey scheme.

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