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Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

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dc.contributor.author Escobar Ruiz, M.A.
dc.contributor.author Subag, E.
dc.contributor.author Miller Jr., W.
dc.date.accessioned 2019-02-19T19:34:18Z
dc.date.available 2019-02-19T19:34:18Z
dc.date.issued 2017
dc.identifier.citation 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. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45
dc.identifier.other DOI:10.3842/SIGMA.2017.099
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/149270
dc.description.abstract 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. uk_UA
dc.description.sponsorship 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]. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title Contractions of Degenerate Quadratic Algebras, Abstract and Geometric uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA


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