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dc.contributor.authorHeinonen, Robinen_NZ
dc.contributor.authorKalnins, Ernie G.en_NZ
dc.contributor.authorMiller, W., Jr.en_NZ
dc.contributor.authorSubag, Eyalen_NZ
dc.date.accessioned2016-04-27T21:51:29Z
dc.date.available2015en_NZ
dc.date.available2016-04-27T21:51:29Z
dc.date.issued2015en_NZ
dc.identifier.citationHeinonen, R., Kalnins, E. G., Miller, W., Jr., & Subag, E. (2015). Structure relations and darboux contractions for 2D 2nd order superintegrable systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 11. http://doi.org/10.3842/SIGMA.2015.043en
dc.identifier.urihttps://hdl.handle.net/10289/10144
dc.description.abstractTwo-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 h → 0 and nonrelativistic phenomena from special relativistic as c → ∞, 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 classif ied 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, C) in flat space and o(3, 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.en_NZ
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherInstitute of Mathematics of National Academy of Science of Ukraine
dc.rightsThe authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License
dc.titleStructure relations and darboux contractions for 2D 2nd order superintegrable systemsen_NZ
dc.typeJournal Article
dc.identifier.doi10.3842/SIGMA.2015.043en_NZ
dc.relation.isPartOfSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)en_NZ
pubs.elements-id128745
pubs.volume11en_NZ
dc.identifier.eissn1815-0659en_NZ
uow.identifier.article-no043


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