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

SIGMA 13 (2017), 013, 38 pages      arXiv:1611.02560      https://doi.org/10.3842/SIGMA.2017.013

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Mauricio A. Escobar Ruiz ab, Ernest G. Kalnins c, Willard Miller Jr. b and Eyal Subag d
a) Instituto de Ciencias Nucleares, UNAM, Apartado Postal 70-543, 04510 Mexico D.F. Mexico
b) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
c) Department of Mathematics, University of Waikato, Hamilton, New Zealand
d) Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, 16802, USA

Received November 19, 2016, in final form February 27, 2017; Published online March 06, 2017

Abstract
Quadratic algebras are generalizations of Lie algebras which 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 Bôcher contractions of the conformal Lie algebra ${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and ${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.

Key words: contractions; quadratic algebras; superintegrable systems; conformal superintegrability.

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