Bôcher and abstract contractions of 2nd order quadratic algebras

Mauricio A. Escobar Ruiz, Ernest G. Kalnins, Willard Miller, Eyal Subag

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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 so(4, 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 e(2, C) and so(3, 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.

Original languageEnglish (US)
Article number013
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
StatePublished - Mar 6 2017

Bibliographical note

Funding Information:
This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller Jr. and by grant (# 250881 to M.A. Escobar-Ruiz). The author M.A. Escobar-Ruiz is grateful to ICN UNAM for the kind hospitality during his visit, where a part of the research was done, he was supported in part by DGAPA grant IN108815 (Mexico).

Publisher Copyright:
© 2017, Institute of Mathematics. All Rights Reserved.


  • Conformal superintegrability
  • Contractions
  • Quadratic algebras
  • Superintegrable systems


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