Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
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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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149270 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric / M.A. Escobar Ruiz, Willard Miller Jr, E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862538460697460736 |
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| author | Escobar Ruiz, M.A. Subag, E. Miller Jr., W. |
| author_facet | Escobar Ruiz, M.A. Subag, E. Miller Jr., W. |
| citation_txt | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric / M.A. Escobar Ruiz, Willard Miller Jr, E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 23 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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between Bôcher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all Bôcher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric Bôcher contractions.
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| first_indexed | 2025-11-24T15:05:00Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149270 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T15:05:00Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | Escobar Ruiz, M.A. Subag, E. Miller Jr., W. 2019-02-19T19:34:18Z 2019-02-19T19:34:18Z 2017 Contractions of Degenerate Quadratic Algebras, Abstract and Geometric / M.A. Escobar Ruiz, Willard Miller Jr, E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 DOI:10.3842/SIGMA.2017.099 https://nasplib.isofts.kiev.ua/handle/123456789/149270 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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between Bôcher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all Bôcher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric Bôcher contractions. 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). The author M.A. Escobar 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). We thank a referee
 for pointing out the relevance of references [4, 6, 20]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Contractions of Degenerate Quadratic Algebras, Abstract and Geometric Article published earlier |
| spellingShingle | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric Escobar Ruiz, M.A. Subag, E. Miller Jr., W. |
| title | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric |
| title_full | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric |
| title_fullStr | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric |
| title_full_unstemmed | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric |
| title_short | Contractions of Degenerate Quadratic Algebras, Abstract and Geometric |
| title_sort | contractions of degenerate quadratic algebras, abstract and geometric |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149270 |
| work_keys_str_mv | AT escobarruizma contractionsofdegeneratequadraticalgebrasabstractandgeometric AT subage contractionsofdegeneratequadraticalgebrasabstractandgeometric AT millerjrw contractionsofdegeneratequadraticalgebrasabstractandgeometric |