Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T06:39:04.649Z Has data issue: false hasContentIssue false
  • English
  • Français

Every symplectic manifold is a (linear) coadjoint orbit

Part of: Symplectic geometry, contact geometry Infinite-dimensional manifolds

Published online by Cambridge University Press:  18 May 2021

Paul Donato  and
Patrick Iglesias-Zemmour
Paul Donato
Affiliation:
Institut de Mathématiques de Marseille, Aix-Marseille Université, 3 Place Victor-Hugo, 13331Marseille Cedex 3, France e-mail: paul.donato@univ-amu.fr
Patrick Iglesias-Zemmour
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 9190401Jerusalem, Israel e-mail: piz@math.huji.ac.ilURL:http://math.huji.ac.il/~piz
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every symplectic manifold is a coadjoint orbit of the group of automorphisms of its integration bundle, acting linearly on its space of momenta, for any group of periods of the symplectic form. This result generalizes the Kirilov–Kostant–Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group and the symplectic form is integral.

Keywords

Diffeologysymplectic geometryquantization

MSC classification

Primary: 53D05: Symplectic manifolds, general
Secondary: 58B99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 2 , June 2022 , pp. 345 - 360
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boothby, W. M., Transitivity of the automorphisms of certain geometric structures. Trans. Amer. Math. Soc. 137(1969), 93–100.CrossRefGoogle Scholar
Donato, P., Revêtement et groupe fondamental des espaces différentiels homogènes. Thèse de doctorat d’état, Université de Provence, Marseille, 1984.Google Scholar
Donato, P. and Iglesias, P., Exemple de groupes différentiels: flots irrationnels sur le tore, Preprint CPT-83/P.1524. Centre de Physique Théorique, Marseille, July 1983. http://math.huji.ac.il/~piz/documents/EDGDFISLT.pdf Google Scholar
https://mathoverflow.net/questions/147395/ Google Scholar
Iglesias, P. and Lachaud, G., Espaces différentiables singuliers et corps de nombres algébriques. Ann. Inst. Fourier 40(1990), no. 1, 723–737.Google Scholar
Iglesias-Zemmour, P., La Trilogie du moment. Ann. Inst. Fourier 45(1995), no. 3, 825–857.CrossRefGoogle Scholar
Iglesias-Zemmour, P., The moment maps in diffeology. Mem. Amer. Math. Soc. 207(2010), no. 972.Google Scholar
Iglesias-Zemmour, P., Diffeology. AMS Mathematical Surveys and Monographs, 185, 2013.CrossRefGoogle Scholar
Iglesias-Zemmour, P., Every symplectic manifold is a coadjoint orbit. Proceedings of Frontiers of Fundamental Physics 14—Proceedings of Sciences (FFP14), vol. 224(2016) pp. 1–17. https://doi.org/10.22323/1.224.0141 CrossRefGoogle Scholar
Iglesias-Zemmour, P., Differential of holonomy for torus bundles. Blog Post (2019). http://math.huji.ac.il/~piz/documents/DBlog-EX-DOHFTB.pdf Google Scholar
Kirillov, A. A., Elements de la théorie des représentations. MIR Ed, Moscow, Russia, 1974.Google Scholar
Kostant, B., Orbits and quantization theory. In: Proceedings of International Congress of Mathematicians, Vol. 2, Nice, France, 1970, pp. 395–400.Google Scholar
Marsden, J. and Weinstein, A., The hamiltonian structure of the maxwell-vlasov equations. Physica D. 4(1982), 394–406.CrossRefGoogle Scholar
Omohundro, S. M., Geometric perturbation theory in physics . World Scientific, 1986. https://doi.org/10.1142/0287 Google Scholar
Souriau, J.-M., Structure des systèmes dynamiques . Dunod Ed, Paris, 1970.Google Scholar