Hofer's metric in compact Lie groups

Autores
Larotonda, Gabriel Andrés; Miglioli, Martín Carlos
Año de publicación
2023
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.
Fil: Larotonda, Gabriel Andrés. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Miglioli, Martín Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Materia
HOFER’S METRIC
COMPACT LIE GROUP
HAMILTONIAN ACTION
MOMENT POLYTOPE
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/212087
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/212087
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Hofer's metric in compact Lie groupsLarotonda, Gabriel AndrésMiglioli, Martín CarlosHOFER’S METRICCOMPACT LIE GROUPHAMILTONIAN ACTIONMOMENT POLYTOPEhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.Fil: Larotonda, Gabriel Andrés. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Miglioli, Martín Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaEuropean Mathematical Society2023-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/212087Larotonda, Gabriel Andrés; Miglioli, Martín Carlos; Hofer's metric in compact Lie groups; European Mathematical Society; Groups Geometry And Dynamics; 17; 3; 5-2023; 839–8981661-7207CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://ems.press/doi/10.4171/GGD/721info:eu-repo/semantics/altIdentifier/doi/10.4171/GGD/721info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1907.09843info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:20:48Zoai:ri.conicet.gov.ar:11336/212087instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-10-22 11:20:48.776CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Hofer's metric in compact Lie groups
title Hofer's metric in compact Lie groups
spellingShingle Hofer's metric in compact Lie groups
Larotonda, Gabriel Andrés
HOFER’S METRIC
COMPACT LIE GROUP
HAMILTONIAN ACTION
MOMENT POLYTOPE
title_short Hofer's metric in compact Lie groups
title_full Hofer's metric in compact Lie groups
title_fullStr Hofer's metric in compact Lie groups
title_full_unstemmed Hofer's metric in compact Lie groups
title_sort Hofer's metric in compact Lie groups
dc.creator.none.fl_str_mv Larotonda, Gabriel Andrés
Miglioli, Martín Carlos
author Larotonda, Gabriel Andrés
author_facet Larotonda, Gabriel Andrés
Miglioli, Martín Carlos
author_role author
author2 Miglioli, Martín Carlos
author2_role author
dc.subject.none.fl_str_mv HOFER’S METRIC
COMPACT LIE GROUP
HAMILTONIAN ACTION
MOMENT POLYTOPE
topic HOFER’S METRIC
COMPACT LIE GROUP
HAMILTONIAN ACTION
MOMENT POLYTOPE
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.
Fil: Larotonda, Gabriel Andrés. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Miglioli, Martín Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
description In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.
publishDate 2023
dc.date.none.fl_str_mv 2023-05
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/212087
Larotonda, Gabriel Andrés; Miglioli, Martín Carlos; Hofer's metric in compact Lie groups; European Mathematical Society; Groups Geometry And Dynamics; 17; 3; 5-2023; 839–898
1661-7207
CONICET Digital
CONICET
url http://hdl.handle.net/11336/212087
identifier_str_mv Larotonda, Gabriel Andrés; Miglioli, Martín Carlos; Hofer's metric in compact Lie groups; European Mathematical Society; Groups Geometry And Dynamics; 17; 3; 5-2023; 839–898
1661-7207
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://ems.press/doi/10.4171/GGD/721
info:eu-repo/semantics/altIdentifier/doi/10.4171/GGD/721
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1907.09843
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv European Mathematical Society
publisher.none.fl_str_mv European Mathematical Society
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.982451

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