On moment maps and Jacobi manifolds

The main goal of this work is to introduce the idea of a Hamiltonian action in the context of Jacobi structures on line bundles. This work aims to make these construction without relying on the "Poissonization trick". Our definition allows us to recover the notion of (weakly)Hamiltonian ac...

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Autores:: Leguizamón Robayo, Alexander

Tipo de recurso:

Fecha de publicación:: 2021

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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dc.title.eng.fl_str_mv	On moment maps and Jacobi manifolds
dc.title.translated.spa.fl_str_mv	Sobre mapas momento y variedades de Jacobi
title	On moment maps and Jacobi manifolds
spellingShingle	On moment maps and Jacobi manifolds 510 - Matemáticas Sistemas Hamiltonianos Hamilton-Jacobi equation Jacobi structures Contact Manifolds Locally conformally symplectic structures Hamiltonian actions Moment Maps Estructuras de Jacobi Variedades de contact Estructuras localmente conformemente simplécticas Acción Hamiltoniana Aplicación momento
title_short	On moment maps and Jacobi manifolds
title_full	On moment maps and Jacobi manifolds
title_fullStr	On moment maps and Jacobi manifolds
title_full_unstemmed	On moment maps and Jacobi manifolds
title_sort	On moment maps and Jacobi manifolds
dc.creator.fl_str_mv	Leguizamón Robayo, Alexander
dc.contributor.advisor.none.fl_str_mv	Sepe, Daniele Martinez Alba, Nicolas
dc.contributor.author.none.fl_str_mv	Leguizamón Robayo, Alexander
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas
topic	510 - Matemáticas Sistemas Hamiltonianos Hamilton-Jacobi equation Jacobi structures Contact Manifolds Locally conformally symplectic structures Hamiltonian actions Moment Maps Estructuras de Jacobi Variedades de contact Estructuras localmente conformemente simplécticas Acción Hamiltoniana Aplicación momento
dc.subject.lemb.none.fl_str_mv	Sistemas Hamiltonianos Hamilton-Jacobi equation
dc.subject.proposal.eng.fl_str_mv	Jacobi structures Contact Manifolds Locally conformally symplectic structures Hamiltonian actions Moment Maps
dc.subject.proposal.spa.fl_str_mv	Estructuras de Jacobi Variedades de contact Estructuras localmente conformemente simplécticas Acción Hamiltoniana Aplicación momento
description	The main goal of this work is to introduce the idea of a Hamiltonian action in the context of Jacobi structures on line bundles. This work aims to make these construction without relying on the "Poissonization trick". Our definition allows us to recover the notion of (weakly)Hamiltonian action in the context of Poisson, contact, and locally conformally symplectic geometry.
publishDate	2021
dc.date.accessioned.none.fl_str_mv	2021-06-21T22:30:00Z
dc.date.available.none.fl_str_mv	2021-06-21T22:30:00Z
dc.date.issued.none.fl_str_mv	2021
dc.type.spa.fl_str_mv	Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv	info:eu-repo/semantics/masterThesis
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dc.identifier.instname.spa.fl_str_mv	Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv	Repositorio Institucional Universidad Nacional de Colombia
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url	https://repositorio.unal.edu.co/handle/unal/79669 https://repositorio.unal.edu.co/
identifier_str_mv	Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv	eng
language	eng
dc.relation.references.spa.fl_str_mv	S. C. Coutinho. A Primer of Algebraic D-Modules. London Mathematical Society Student Texts. Cambridge: Cambridge University Press, 1995. isbn: 978-0-521-55119-9. doi: 10 . 1017/CBO9780511623653. url: https://www.cambridge.org/core/books/primer- of- algebraic-dmodules/87B8F8AB3B53DBA8A8BD33A058E54473. Marius Crainic. Mastermath course Differential Geometry 2015/2016. Lecture Notes. Uni- versity of Utrecht, 2016. Marius Crainic and Marı́a Amelia Salazar. “Jacobi structures and Spencer operators”. In: Journal des Mathematiques Pures et Appliquees 103.2 (2015), pp. 504–521. doi: 10.1016/ j.matpur.2014.04.012. Marius Crainic and Chenchang Zhu. “Integrability of Jacobi and Poison Structures”. In: Annales de L’Institut Fourier 57.4 (2007), pp. 1181–1216. url: http://aif.cedram.org/ item?id=AIF_2007__57_4_1181_0. Pierre Dazord, André Lichnerowicz, and Charles-Michel Marle. “Structure locale des variétés de Jacobi”. In: Journal de mathématiques pures et appliquées 70.1 (1991), pp. 101–152. Rui Loja Fernandes. Differential Geometry. Lecture Notes. University of Illinois Urbana Champaign, Oct. 2020. url: https : / / faculty . math . illinois . edu / ~ruiloja / Meus - papers/HTML/notesDG.pdf. Rui Loja Fernandes and Ioan Marcut. Lectures on Poisson geometry. 2015. url: https : //faculty.math.illinois.edu/~ruiloja/Math595/Spring14/book.pdf. Hansjörg Geiges. An Introduction to Contact Topology. Cambridge University Press, 2008. Hansjörg Geiges. “Contact geometry”. In: Handbook of Differential Geometry Vol. 2. July 2006, pp. 315–312. doi: 10.1007/978-94-011-3330-2_3. url: http://arxiv.org/abs/ math/0307242. Janusz Grabowski. “Local lie algebra determines base manifold”. In: Progress in Mathematics 252.2 (2007), pp. 131–145. doi: 10.1007/978-0-8176-4530-4_9. A A Kirillov. “Local Lie Algebras”. In: Russian Mathematical Surveys 31.4 (Aug. 1976), pp. 55–75. doi: 10.1070/RM1976v031n04ABEH001556. url: http://stacks.iop.org/0036- 0279/31/i=4/a=R02?key=crossref.b88697572b48af4d7d2216e575131451. Ivan Kolář, Peter W. Michor, and Jan Slovák. Natural operations in differential geometry. en. Web Version, 1st Ed. 1993. Berlin ; New York: Springer-Verlag, 2005. isbn: 978-3-540-56235-1 978-0-387-56235-3. url: https://www.emis.de/monographs/KSM/kmsbookh.pdf. Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke. Poisson Structures. 1st ed. Publication Title: Grundlehren der mathematischen Wissenschaften. Basel: Springer-Verlag Berlin Heidelberg, 2013. isbn: 978-3-642-31090-4. doi: 10.1007/978-3-642-31090-4. url: https://www.springer.com/gp/book/9783642310898. John Lee. Introduction to Smooth Manifolds. 2nd ed. Pages: XVI, 708. New York: Springer- Verlag New York, 2012. isbn: 978-1-4419-9981-8. André Lichnerowicz. “Les variétés de Jacobi et leurs algébres de Lie associées”. In: J. Math. Pures Appl 57 (1978), pp. 453–488. Frank Loose. “Reduction in Contact Geometry”. In: Journal of Lie Theory 11.1 (2001), pp. 9–22. url: https://www.emis.de/journals/JLT/vol.11_no.1/2.html (visited on 11/01/2020). Charles-Michel Marle. “On Jacobi Manifolds and Jacobi Bundles”. In: In Dazord P., We- instein A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Ed. by Mathe- matical Sciences Research Institute Publications. Vol. 111. New York, NY: Springer, New York, NY, 1991, pp. 1009–1010. doi: 10 . 1007 / 978 - 1 - 4613 - 9719 - 9 _ 16. url: http : //link.springer.com/10.1007/978-1-4613-9719-9_16. Jet Nestruev. Smooth manifolds and observables. English. OCLC: 1199307234. 2020. isbn: 978-3-030-45649-8. Marı́a Amelia Salazar and Daniele Sepe. “Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators”. In: 13 (2014). doi: 10.3842/SIGMA.2017.033. url: http://arxiv. org/abs/1406.2138%0Ahttp://dx.doi.org/10.3842/SIGMA.2017.033. Daniele Sepe. Geometria Simpletica. Universidade Federal Fluminense (UFF), 2020. Ana Canas da Silva. Lectures on Symplectic Geometry. Vol. 1764. Lecture Notes in Math- ematics. Publication Title: Lectures on Symplectic Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. isbn: 978-3-540-42195-5. doi: 10.1007/b80865. url: http://link. springer.com/10.1007/978-3-540-45330-7. Ana Cannas da Silva and Alan Weinstein. Geometric Models for Noncommutative Algebras. Berkeley, CA: American Mathematical Society, 2000. url: https://bookstore.ams.org/ bmln-10 (visited on 10/19/2020). Miron Stanciu. “Locally conformally symplectic reduction”. en. In: (Aug. 2018). url: https: //arxiv.org/abs/1809.00034v2 (visited on 10/14/2020). Alfonso Giuseppe Tortorella. “Deformations of coisotropic submanifolds in Jacobi manifolds”. In: arXiv:1705.08962 [math] (May 2017). arXiv: 1705.08962. url: http://arxiv.org/abs/ 1705.08962 (visited on 07/22/2020). Izu Vaisman. “Locally Conformal Symplectic Manifolds”. In: International Journal of Math- ematics and Mathematical Sciences 8 (Jan. 1985). doi: 10.1155/S0161171285000564. Luca Vitagliano. “Dirac–Jacobi bundles”. In: Journal of Symplectic Geometry 16.2 (2018), pp. 485–561. doi: 10.4310/JSG.2018.v16.n2.a4. url: http://www.intlpress.com/site/ pub/pages/journals/items/jsg/content/vols/0016/0002/a004/. Bibliography Luca Vitagliano and Aı̈ssa Wade. “Holomorphic Jacobi Manifolds and Holomorphic Contact Groupoids”. en. In: Math. Z. 294.3-4 (2020). arXiv: 1710.03300, pp. 1181–1225. issn: 0025- 5874, 1432-1823. doi: 10.1007/s00209-019-02320-x. url: http://arxiv.org/abs/1710. 03300 (visited on 07/20/2020). Carlos Zapata-Carratala. “A Landscape of Hamiltonian Phase Spaces: on the foundations and generalizations of one of the most powerful ideas of modern science”. PhD thesis. Oct. 2019. url: http://arxiv.org/abs/1910.08469.
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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv	Bogotá - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisher.department.spa.fl_str_mv	Departamento de Matemáticas
dc.publisher.faculty.spa.fl_str_mv	Facultad de Ciencias
dc.publisher.place.spa.fl_str_mv	Bogotá, Colombia
dc.publisher.branch.spa.fl_str_mv	Universidad Nacional de Colombia - Sede Bogotá
institution	Universidad Nacional de Colombia
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spelling	Atribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Sepe, Danieled40a0bdc3d5d3fe10adc8702ae095999Martinez Alba, Nicolas1f63b38dcb81b6cd934eaffcd285b12aLeguizamón Robayo, Alexanderbdc73b2ac8bea0faa96c6bfbc02c5f4b2021-06-21T22:30:00Z2021-06-21T22:30:00Z2021https://repositorio.unal.edu.co/handle/unal/79669Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/The main goal of this work is to introduce the idea of a Hamiltonian action in the context of Jacobi structures on line bundles. This work aims to make these construction without relying on the "Poissonization trick". Our definition allows us to recover the notion of (weakly)Hamiltonian action in the context of Poisson, contact, and locally conformally symplectic geometry.En el siguiente trabajo introducimos la idea de acción Hamiltoniana en el contexto de la geometría de Jacobi en fibrados de línea generales. Esta construcción la realizamos de forma intrínseca sin necesidad de recurrir al ”truco de Poissonización”. El concepto de acción Hamiltoniana en geometría de Jacobi nos permite recuperar resultados conocidos en geometría de Poisson, contacto, y localmente conformemente simplécticaMaestríaMagíster en Ciencias - MatemáticasGeometría Diferencial103 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - MatemáticasSistemas HamiltonianosHamilton-Jacobi equationJacobi structuresContact ManifoldsLocally conformally symplectic structuresHamiltonian actionsMoment MapsEstructuras de JacobiVariedades de contactEstructuras localmente conformemente simplécticasAcción HamiltonianaAplicación momentoOn moment maps and Jacobi manifoldsSobre mapas momento y variedades de JacobiTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMS. C. Coutinho. A Primer of Algebraic D-Modules. London Mathematical Society Student Texts. Cambridge: Cambridge University Press, 1995. isbn: 978-0-521-55119-9. doi: 10 . 1017/CBO9780511623653. url: https://www.cambridge.org/core/books/primer- of- algebraic-dmodules/87B8F8AB3B53DBA8A8BD33A058E54473.Marius Crainic. Mastermath course Differential Geometry 2015/2016. Lecture Notes. Uni- versity of Utrecht, 2016.Marius Crainic and Marı́a Amelia Salazar. “Jacobi structures and Spencer operators”. In: Journal des Mathematiques Pures et Appliquees 103.2 (2015), pp. 504–521. doi: 10.1016/ j.matpur.2014.04.012.Marius Crainic and Chenchang Zhu. “Integrability of Jacobi and Poison Structures”. In: Annales de L’Institut Fourier 57.4 (2007), pp. 1181–1216. url: http://aif.cedram.org/ item?id=AIF_2007__57_4_1181_0.Pierre Dazord, André Lichnerowicz, and Charles-Michel Marle. “Structure locale des variétés de Jacobi”. In: Journal de mathématiques pures et appliquées 70.1 (1991), pp. 101–152.Rui Loja Fernandes. Differential Geometry. Lecture Notes. University of Illinois Urbana Champaign, Oct. 2020. url: https : / / faculty . math . illinois . edu / ~ruiloja / Meus - papers/HTML/notesDG.pdf.Rui Loja Fernandes and Ioan Marcut. Lectures on Poisson geometry. 2015. url: https : //faculty.math.illinois.edu/~ruiloja/Math595/Spring14/book.pdf.Hansjörg Geiges. An Introduction to Contact Topology. Cambridge University Press, 2008.Hansjörg Geiges. “Contact geometry”. In: Handbook of Differential Geometry Vol. 2. July 2006, pp. 315–312. doi: 10.1007/978-94-011-3330-2_3. url: http://arxiv.org/abs/ math/0307242.Janusz Grabowski. “Local lie algebra determines base manifold”. In: Progress in Mathematics 252.2 (2007), pp. 131–145. doi: 10.1007/978-0-8176-4530-4_9.A A Kirillov. “Local Lie Algebras”. In: Russian Mathematical Surveys 31.4 (Aug. 1976), pp. 55–75. doi: 10.1070/RM1976v031n04ABEH001556. url: http://stacks.iop.org/0036- 0279/31/i=4/a=R02?key=crossref.b88697572b48af4d7d2216e575131451.Ivan Kolář, Peter W. Michor, and Jan Slovák. Natural operations in differential geometry. en. Web Version, 1st Ed. 1993. Berlin ; New York: Springer-Verlag, 2005. isbn: 978-3-540-56235-1 978-0-387-56235-3. url: https://www.emis.de/monographs/KSM/kmsbookh.pdf.Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke. Poisson Structures. 1st ed. Publication Title: Grundlehren der mathematischen Wissenschaften. Basel: Springer-Verlag Berlin Heidelberg, 2013. isbn: 978-3-642-31090-4. doi: 10.1007/978-3-642-31090-4. url: https://www.springer.com/gp/book/9783642310898.John Lee. Introduction to Smooth Manifolds. 2nd ed. Pages: XVI, 708. New York: Springer- Verlag New York, 2012. isbn: 978-1-4419-9981-8.André Lichnerowicz. “Les variétés de Jacobi et leurs algébres de Lie associées”. In: J. Math. Pures Appl 57 (1978), pp. 453–488.Frank Loose. “Reduction in Contact Geometry”. In: Journal of Lie Theory 11.1 (2001), pp. 9–22. url: https://www.emis.de/journals/JLT/vol.11_no.1/2.html (visited on 11/01/2020).Charles-Michel Marle. “On Jacobi Manifolds and Jacobi Bundles”. In: In Dazord P., We- instein A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Ed. by Mathe- matical Sciences Research Institute Publications. Vol. 111. New York, NY: Springer, New York, NY, 1991, pp. 1009–1010. doi: 10 . 1007 / 978 - 1 - 4613 - 9719 - 9 _ 16. url: http : //link.springer.com/10.1007/978-1-4613-9719-9_16.Jet Nestruev. Smooth manifolds and observables. English. OCLC: 1199307234. 2020. isbn: 978-3-030-45649-8.Marı́a Amelia Salazar and Daniele Sepe. “Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators”. In: 13 (2014). doi: 10.3842/SIGMA.2017.033. url: http://arxiv. org/abs/1406.2138%0Ahttp://dx.doi.org/10.3842/SIGMA.2017.033.Daniele Sepe. Geometria Simpletica. Universidade Federal Fluminense (UFF), 2020.Ana Canas da Silva. Lectures on Symplectic Geometry. Vol. 1764. Lecture Notes in Math- ematics. Publication Title: Lectures on Symplectic Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. isbn: 978-3-540-42195-5. doi: 10.1007/b80865. url: http://link. springer.com/10.1007/978-3-540-45330-7.Ana Cannas da Silva and Alan Weinstein. Geometric Models for Noncommutative Algebras. Berkeley, CA: American Mathematical Society, 2000. url: https://bookstore.ams.org/ bmln-10 (visited on 10/19/2020).Miron Stanciu. “Locally conformally symplectic reduction”. en. In: (Aug. 2018). url: https: //arxiv.org/abs/1809.00034v2 (visited on 10/14/2020).Alfonso Giuseppe Tortorella. “Deformations of coisotropic submanifolds in Jacobi manifolds”. In: arXiv:1705.08962 [math] (May 2017). arXiv: 1705.08962. url: http://arxiv.org/abs/ 1705.08962 (visited on 07/22/2020).Izu Vaisman. “Locally Conformal Symplectic Manifolds”. In: International Journal of Math- ematics and Mathematical Sciences 8 (Jan. 1985). doi: 10.1155/S0161171285000564.Luca Vitagliano. “Dirac–Jacobi bundles”. In: Journal of Symplectic Geometry 16.2 (2018), pp. 485–561. doi: 10.4310/JSG.2018.v16.n2.a4. url: http://www.intlpress.com/site/ pub/pages/journals/items/jsg/content/vols/0016/0002/a004/. BibliographyLuca Vitagliano and Aı̈ssa Wade. “Holomorphic Jacobi Manifolds and Holomorphic Contact Groupoids”. en. In: Math. Z. 294.3-4 (2020). arXiv: 1710.03300, pp. 1181–1225. issn: 0025- 5874, 1432-1823. doi: 10.1007/s00209-019-02320-x. url: http://arxiv.org/abs/1710. 03300 (visited on 07/20/2020).Carlos Zapata-Carratala. “A Landscape of Hamiltonian Phase Spaces: on the foundations and generalizations of one of the most powerful ideas of modern science”. PhD thesis. Oct. 2019. url: http://arxiv.org/abs/1910.08469.GeneralLICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/79669/1/license.txtcccfe52f796b7c63423298c2d3365fc6MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8914https://repositorio.unal.edu.co/bitstream/unal/79669/3/license_rdf24013099e9e6abb1575dc6ce0855efd5MD53ORIGINALEncabezadoTesisMSc.pdfEncabezadoTesisMSc.pdfTesis de Maestría en Ciencias - Matemáticasapplication/pdf703143https://repositorio.unal.edu.co/bitstream/unal/79669/2/EncabezadoTesisMSc.pdf708f7cf8b3f4e80c0422e3139a984d84MD52THUMBNAILEncabezadoTesisMSc.pdf.jpgEncabezadoTesisMSc.pdf.jpgGenerated Thumbnailimage/jpeg3565https://repositorio.unal.edu.co/bitstream/unal/79669/4/EncabezadoTesisMSc.pdf.jpgfbe88bcb98e669cf5654e251640f95baMD54unal/79669oai:repositorio.unal.edu.co:unal/796692024-07-22 00:43:08.717Repositorio Institucional Universidad Nacional de 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On moment maps and Jacobi manifolds

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