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Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
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.
Type
Journal Article
Type of thesis
Series
Citation
Escobar Ruiz, M. A., Kalnins, E. G., Miller, W., Jr., & Subag, E. (2017). Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras. Symmetry, Integrability and Geometry: Methods and Applications, 13. https://doi.org/10.3842/SIGMA.2017.013
Date
2017
Publisher
Institute of Mathematics of NAS of Ukraine
Degree
Supervisors
Rights
© 2017 copyright with the authors. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.