Contractions of degenerate quadratic Algebras, abstract and geometric

Mauricio A. Escobar Ruiz, Willard Miller, Eyal Subag

Research output: Contribution to journalArticle

Abstract

uadratic 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.

Original languageEnglish (US)
Article number099
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume13
DOIs
StatePublished - Dec 31 2017

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Quadratic Algebras
Contraction
Lie Algebra
Conformal Algebra
Algebra
Free Algebras
Classical Mechanics
Canonical form
Quantum Mechanics
Exception
Phase Space
Curvature
Classify
Distinct
Symmetry

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

Cite this

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abstract = "uadratic 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 B{\^o}cher 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 B{\^o}cher 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 B{\^o}cher 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 B{\^o}cher contractions.",
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Contractions of degenerate quadratic Algebras, abstract and geometric. / Escobar Ruiz, Mauricio A.; Miller, Willard; Subag, Eyal.

In: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), Vol. 13, 099, 31.12.2017.

Research output: Contribution to journalArticle

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