Actions symplectiques de groupes compacts

Benoist, Yves

doi:10.1007/s10711-007-9124-1

Cited by 10 publications

(26 citation statements)

References 3 publications

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“…This result motivated many works, but no attention has been given to the interesting case of non-Hamiltonian actions. (In circulating this work we have been told that our Theorem 3.16 was also announced in the communication [4] and in the article [5] by Benoist. ) The relevance of non-Hamiltonian actions is well known to physics community, beginning with Novikov's observations in [16].…”

Section: Introductionmentioning

confidence: 89%

Convexity of Multi-valued Momentum Maps

Giacobbe

2005

Geom Dedicata

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A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian. (2000). 37Jxx, 37Kxx. Mathematics Subject Classifications

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Section: Introductionmentioning

confidence: 89%

Convexity of Multi-valued Momentum Maps

Giacobbe

2005

Geom Dedicata

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“…Other convexity theorems were proven later by Birtea-Ortega-Ratiu [20], Kirwan [93] (in the case of compact, non-abelian group actions), Benoist [15], and Giacobbe [58], to name a few. Convexity in the case of Poisson actions has been studied by Alekseev, Flaschka-Ratiu, Ortega-Ratiu and Weinstein [4,47,118,158] among others.…”

Section: 2mentioning

confidence: 99%

“…defined on the manifold. For instance in the case of symplectic group actions which we will discuss later the local normal form of symplectic Hamiltonian actions is due to Guillemin-Marle-Sternberg [74,105], and in the general case of symplectic actions to Ortega-Ratiu [117] and Benoist [15], in a neighborhood of an orbit.…”

Section: 2mentioning

confidence: 99%

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Hamiltonian and symplectic symmetries: An introduction

Pelayo

2017

Bull. Amer. Math. Soc.

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Abstract. Classical mechanical systems are modeled by a symplectic manifold (M, ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finitedimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent results on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry.

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“…Puisque, d'après la Proposition 3.1, N est un espace hamiltonien au sens usuel pour l'action du tore T , on est ramené au théorème d'Atiyah-Guillemin-Sternberg. En réalité, la preuve est techniquement plus subtile car l'application moment μ| N n'est pas nécessairement propre, mais on peut lever ces difficultés de nature topologique grâce à l'application du principe local-global (voir [9]), que nous ne détaillerons pas ici (voir aussi [3]). …”

Section: Existence D'une Tranche Symplectiqueunclassified

Un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie

Schaffhauser

2007

Comptes Rendus. Mathématique

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Reçu le 14 août 2006 ; accepté après révision le 29 mai 2007 Présenté par Michel Duflo RésuméDans cette Note, nous présentons l'énoncé et les principales idées de la démonstration d'un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie. Ce résultat est un analogue quasi-hamiltonien du théorème de O'Shea et Sjamaar dans le cadre hamiltonien usuel. On démontre que l'image par l'application moment du lieu des points fixes d'une involution renversant la 2-forme de structure d'un espace quasi-hamiltonien est un polytope convexe, et l'on décrit ce polytope comme sous-polytope du polytope moment. Pour citer cet article : F. Schaffhauser, C. R. Acad. Sci. Paris, Ser. I 345 (2007). AbstractA real convexity theorem for group-valued momentum maps. In this Note, we state and give the main ideas of the proof of a real convexity theorem for group-valued momentum maps. This result is a quasi-Hamiltonian analogue of the O'Shea-Sjamaar theorem in the usual Hamiltonian setting. We prove here that the image under the momentum map of the fixed-point set of a form-reversing involution defined on a quasi-Hamiltonian space is a convex polytope, that we describe as a subpolytope of the momentum polytope. To cite this article: F. Schaffhauser, C. R. Acad. Sci. Paris, Ser. I 345 (2007). Abridged English versionLet (G, (. | .)) be a compact connected Lie group whose Lie algebra g := Lie(G) is endowed with an Adinvariant (positive definite) scalar product (. | .). We denote by θ L = g −1 .dg and by θ R = dg.g −1 the Maurer-Cartan 1-forms on G and by χ the Cartan 3-form, that is to say the bi-invariant 3-form defined for X, Y, Z ∈ g = T 1 G byx) the fundamental vector field associated to X ∈ g by the action of G on M. A quasi-Hamiltonian G-space (M, ω, μ : M → U) is a manifold M, acted upon by the group G, endowed with an invariant 2-form ω such that there exists an equivariant map μ : M → G (for the conjugacy action of G on itself) satisfying:(iii) for all X ∈ g, the interior product ι X ω satisfies ι X ω = 1 2 μ * (θ L + θ R | X). This notion was introduced by Alekseev, Malkin and Meinrenken in [1]and basic examples include the conjugacy classes of the Lie group G. The map μ is called the momentum map, underlining the important analogies with usual Hamiltonian spaces that one encounters in the study of quasi-Hamiltonian spaces. In the rest of this paper, the compact connected Lie group G will be assumed to be simply connected as well. In [1], Alekseev, Malkin and Meinrenken showed that in this case the intersection of the image μ(M) of the momentum map μ with a fundamental domain exp(W) for the conjugacy action (W ⊂ g is a Weyl alcove in the Lie algebra of G) is a convex polytope (a result which they derived from Meinrenken and Woodward in [12]). Here, we study the image μ(M β ) under the momentum map of the fixed-point set M β of an involution β defined on M satisfying β * ω = −ω (we shall say that β reverses the structural 2-form ω) and additional compatibility relations with the action and the moment...

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Actions symplectiques de groupes compacts

Cited by 10 publications

References 3 publications

Convexity of Multi-valued Momentum Maps

Convexity of Multi-valued Momentum Maps

Hamiltonian and symplectic symmetries: An introduction

Un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie

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