Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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Інститут математики НАН України
2017
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| Zitieren: | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862632104904359936 |
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| author | Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. |
| author_facet | Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. |
| citation_txt | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-30T12:16:58Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148617 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T12:16:58Z |
| publishDate | 2017 |
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| spelling | Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. 2019-02-18T16:42:19Z 2019-02-18T16:42:19Z 2017 Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 DOI:10.3842/SIGMA.2017.013 https://nasplib.isofts.kiev.ua/handle/123456789/148617 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. This work was partially supported by a grant from the Simons Foundation (# 208754 to
 Willard Miller Jr. and by CONACYT 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). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Article published earlier |
| spellingShingle | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. |
| title | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras |
| title_full | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras |
| title_fullStr | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras |
| title_full_unstemmed | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras |
| title_short | Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras |
| title_sort | bôcher and abstract contractions of 2nd order quadratic algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148617 |
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