Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
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 superintegra...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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Інститут математики НАН України
2015
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| Cite this: | Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. |
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Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. 2019-02-13T16:29:39Z 2019-02-13T16:29:39Z 2015 Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60 DOI:10.3842/SIGMA.2015.043 https://nasplib.isofts.kiev.ua/handle/123456789/147106 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 ℏ→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 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,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. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html. This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller, Jr.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| spellingShingle |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. |
| title_short |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_full |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_fullStr |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_full_unstemmed |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_sort |
structure relations and darboux contractions for 2d 2nd order superintegrable systems |
| author |
Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. |
| author_facet |
Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 ℏ→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 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,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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147106 |
| citation_txt |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. |
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2025-12-07T18:36:56Z |
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2025-12-07T18:36:56Z |
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1850875698164531200 |