A geometric approach to Lie systems: formalism of Poisson-Hopf deformations

The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite...

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Detalles Bibliográficos
Autor: Fernández Saiz, Eduardo
Tipo de recurso: tesis doctoral
Fecha de publicación:2021
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/11559
Acceso en línea:https://hdl.handle.net/20.500.14352/11559
Access Level:acceso abierto
Palabra clave:512.554.3(043.2)
Lie algebras
Álgebras de Lie
Álgebra
1201 Álgebra
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repository_id_str
spelling A geometric approach to Lie systems: formalism of Poisson-Hopf deformationsUn enfoque geométrico a los sistemas de Lie : formalismo de las deformaciones de álgebras de Poisson–HopfFernández Saiz, Eduardo512.554.3(043.2)Lie algebrasÁlgebras de LieÁlgebra1201 ÁlgebraThe notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, under quantum deformations, Lie systems are extended to generalized systems described by involutive distributions. As a consequence, a quantum deformed Lie system no longer has an underlying Vessiot–Guldberg Lie algebra or a quantum algebra one, but keeps a Poisson–Hopf algebra structure that enables us to obtain, in an explicit way, the t-independent constants of the motion from quantum deformed Casimir invariants, which are potentially useful in a further construction of the generalized notion of superposition rules. We illustrate this approach by considering the non-standard quantum deformation of sl(2) applied to well-known Lie systems, such as the oscillator problem or Milne–Pinney equation, as well as several types of Riccati equations. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass...Universidad Complutense de MadridCampoamor Stursberg, Otto-RudwigHerranz Zorilla, Francisco JoséUniversidad Complutense de Madrid20212021-05-0720212021-05-07doctoral thesishttp://purl.org/coar/resource_type/c_db06info:eu-repo/semantics/doctoralThesisapplication/pdfhttps://hdl.handle.net/20.500.14352/11559reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/115592026-06-02T12:44:21Z
dc.title.none.fl_str_mv A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
Un enfoque geométrico a los sistemas de Lie : formalismo de las deformaciones de álgebras de Poisson–Hopf
title A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
spellingShingle A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
Fernández Saiz, Eduardo
512.554.3(043.2)
Lie algebras
Álgebras de Lie
Álgebra
1201 Álgebra
title_short A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
title_full A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
title_fullStr A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
title_full_unstemmed A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
title_sort A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
dc.creator.none.fl_str_mv Fernández Saiz, Eduardo
author Fernández Saiz, Eduardo
author_facet Fernández Saiz, Eduardo
author_role author
dc.contributor.none.fl_str_mv Campoamor Stursberg, Otto-Rudwig
Herranz Zorilla, Francisco José
Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 512.554.3(043.2)
Lie algebras
Álgebras de Lie
Álgebra
1201 Álgebra
topic 512.554.3(043.2)
Lie algebras
Álgebras de Lie
Álgebra
1201 Álgebra
description The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, under quantum deformations, Lie systems are extended to generalized systems described by involutive distributions. As a consequence, a quantum deformed Lie system no longer has an underlying Vessiot–Guldberg Lie algebra or a quantum algebra one, but keeps a Poisson–Hopf algebra structure that enables us to obtain, in an explicit way, the t-independent constants of the motion from quantum deformed Casimir invariants, which are potentially useful in a further construction of the generalized notion of superposition rules. We illustrate this approach by considering the non-standard quantum deformation of sl(2) applied to well-known Lie systems, such as the oscillator problem or Milne–Pinney equation, as well as several types of Riccati equations. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass...
publishDate 2021
dc.date.none.fl_str_mv 2021
2021-05-07
2021
2021-05-07
dc.type.none.fl_str_mv doctoral thesis
http://purl.org/coar/resource_type/c_db06
dc.type.openaire.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/11559
url https://hdl.handle.net/20.500.14352/11559
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universidad Complutense de Madrid
publisher.none.fl_str_mv Universidad Complutense de Madrid
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301603

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