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Frontmatter

Groupoïdes de Lie et Groupoïdes Symplectiques

Résumé
Le but de cet exposé est de donner une approche géométrique de la théorie des groupoïdes de Lie, approche qui s’avère particulièrement utile dans l’étude des groupoïdes symplectiques. Il s’agit d’énoncer brièvement les résultats de [1].
Claude Albert, Pierre Dazord

Géometrie Globale des Systèmes Hamiltoniens Complètement Intégrables et Variables Action-Angle avec Singularités

Résumé
Toutes les structures considérées dans ce travail sont de classe C . Précisons tout d’abord la notion de complète intégrabilité utilisée ici. Rappelons qu’un système hamiltonien (M 2 n , ω, H) est dit complètement intégrable au sens d’Arnold-Liouville s’il existe un n-uple F = (f 1,..., f n )d’intégrales premières en involution dont les différentielles sont générique-ment indépendantes. Le théorème d’Arnold-Liouville affirme alors que les fibres régulières, compactes et connexes, de F, sont des tores Lagrangiens, et qu’au voisinage de chacun d’eux il existe un système de coordonnées canoniques (q 1,...,q n , θ 1,..., θ n ),dites coordonnées action-angle, où les coordonnées action (q 1,..., q n ) sont à valeurs dans un ouvert de n et les coordonnées angle (θ 1,..., θ n ) à valeurs dans le tore T n ,de manière que f 1,..., f n ne sont fonctions que des variables action. Il en résulte en particulier que le flot du champ hamiltonien X H est quasi-périodique sur ces tores Lagrangiens.
Mohamed Boucetta

Sur Quelques Questions de Géométrie Symplectique

Abstract
This paper summarizes a talk that I gave at the Mathematical Science Research Institute (Berkeley) in June 1989. I consider G-homogeneous symplectic manifolds (M, ω) where G is a solvable Lie group. When the symplectic action G × MM is “regular” and “closed” I sketch the proof of two main results:
(1)
the manifold M has an affinely flat structure (M, D) which preserves a bilagrangian structure on (M, ω) and satisfies the condition that Dω = 0;
 
(2)
the symplectic manifold (M, ω) is a graded symplectic manifold.
 
Nguiffo B. Boyom

Intégration Symplectique des Variétés de Poisson Totalement Asphériques

Abstract
Poisson structures are contravariant structures. Nevertheless there is a good description of regular Poisson manifolds by means of foliated symplectic forms. This point of view makes it easy to lift these structures to the holonomy or homotopy groupoïd of their symplectic (regular) foliation; defining the Poisson realization of the Poisson structure.
The second aim of the paper is to construct the universal symplectic integration of totally aspherical Poisson structures that is regular Poisson structures such that:
i)
the second homotopy group of any symplectic leaf is trivial;
 
ii)
any vanishing cycle is trivial.
 
The universal symplectic integration is a symplectic groupoïd with connected and simply connected fibres which realizes the given Poisson structures.
This construction generalizes the construction of the simply connected Lie group of a given finite-dimensional Lie algebra.
Pierre Dazord, Gilbert Hector

La Première Classe de Chern Comme Obstruction à la Quantification Asymptotique

Résumé
Notre travail trouve son origine dans un article de Karašev et Maslov sur la quantification d’une variété symplectique générale [16]. Cet article pose de nombreux problèmes et contient plusieurs points obscurs, que nous clarifions, ce qui nous permet de répondre positivement à certaines conjectures.
P. Dazord, G. Patissier

Groupes de Poisson Affines

Résumé
Le but de cet article est de présenter une extension naturelle de la notion de groupe de Poisson due à Drinfel’d [5]: la notion de groupe de Poisson affine Cette extension contient toutes les structures de Poisson usuellement introduites sur les groupes, et en particulier les structures de groupes de Poisson, les structures de Poisson invariantes à gauche ou à droite, les structures affines de J.M. Souriau [17] sur les duaux d’algèbres de Lie.
Pierre Dazord, D. Sondaz

Singular Lagrangian Foliation Associated to an Integrable Hamiltonian Vector Field

Abstract
In this paper we show what the geometry of an integrable hamiltonian system is under a rather “generic assumptions”. These hypotheses are closely related to those of Fomenko [10] and [11] on Bott integrals, but are distinct and allow us to study higher codimension singularities. In a “companion” paper Jair Koiller shows this gives a good setting in order to study a perturbed system by Melnikov method. The author thanks the referee for his corrections both mathematical and linguistic.
Nicole Desolneux-Moulis

Hyperbolic Actions of R p on Poisson Manifolds

Abstract
Unless otherwise explicitly stated all manifolds and mappings are C Recall that a Poisson manifold ([W]) is a manifold V with a Lie algebra structure (f,g) ↦ {f,g} on C (V) (the set of C mappings f: VR) such that
$$\{ f,gh\} = \{ f,g\} h + g\{ f,h\}$$
Jean-Paul Dufour

Compactification d’Actions de ℝ n et Variables Action-Angle avec Singularités

Abstract. Résumé
On considère une action infinitésimale de ℝ n sur une variété V, munie d’un espace vectoriel d’intégrales premières; on donne une condition suffisante pour que, au voisinage d’un orbite compacte, il existe une action du tore T n ayant les mème orbites, et commutant avec l’action infinitésimale donnée. Comme corollaire, on retrouve le théorème de H. Eliasson sur l’existence de variables Action-Angle avec singularités pour un système hamiltonien.
J. P. Dufour, P. Molino

On the Diameter of the Symplectomorphism Group of the Ball

Abstract
It is shown that the diameter of the symplectomorphism group of the ball in ℝ2n is infinite.
Yakov Eliashberg, Tudor Ratiu

A Symplectic Analogue of the Mostow-Palais Theorem

Abstract
We show that given a Hamiltonian action of a compact and connected Lie group G on a symplectic manifold (M, ω) of finite type, there exists a linear symplectic action of G on some R 2n equipped with its standard symplectic structure such that (M, ω, G) can be realized as a reduction of this R 2n with the induced action of G.
M. J. Gotay, G. M. Tuynman

Melnikov Formulas For Nearly Integrable Hamiltonian Systems

Abstract
An “intrinsic” Melnikov vector valued function is given, which can be used to detect homoclinic orbits in Hamiltonian perturbations of completely integrable systems. We use the description given by Prof. Nicole Desolneux-Moulis [1] of the dynamics along a singular leaf of the unperturbed system. As an example, it is shown that perturbations of the spherical pendulum on a rotating frame (or in a magnetic field) produce Silnikov’s spiralling chaos.
Jair Koiller

A Non-Linear Hadamard Theorem

Abstract
Using Gromov theory of pseudo-holomorphic curves, we derive a pseudo-holomorphic version of the classical result of Hadamard: a holomorphic function with bounded real part is constant. It is a pleasure to thank Gilbert Hector for providing a much simpler proof of Proposition 1, Michel N’Guiffo Boyom and the referee for valuable remarks.
Jacques Lafontaine

Equivariant Prequantization

Abstract
If (S, ω) is a symplectic manifold, a prequantization of S is a principal circle bundle over S together with a connection form whose curvature is −ω. Such a circle bundle exists iff the period group of ω is contained in ℤ; i.e., the class [ω] ∈ H 2(S, ℝ) comes from an integral class. If S is simply connected it follows from the universal coefficient theorem that the integral class is unique. Also note that for simply connected S, the period group of ω is in ℤ iff the spherical period group is in ℤ; i.e., π 2(S) ≅ H 2(S).1 If S is not simply connected it may have inequivalent prequantizations. Inequivalent prequantizations of S also induce inequivalent prequantizations of S × \(\bar S\), \(\bar S\) denoting (S, −ω). But one can show such prequantizations become equivalent when pulled back to the fundamental groupoid (\(\pi \left( S \right) = \tilde S \times \tilde S/{\pi _1}\left( S \right),\lambda :\tilde S \to S\) the universal cover, with the form induced from S × \(\bar S\) by λ × λ). Further if we only assume ω is integral on spherical classes, no prequantization may exist. In his preprint [10], Alan Weinstein gives a method for prequantizing the fundamental groupoid of a symplectic manifold (S, ω) when ω is integral on spherical classes, using connection theory. His result is equivalent to the statement (Corollary 1.3): For any symplectic manifold (S, ω) the period group of the fundamental groupoid π(S) is contained in ℤ iff the spherical period group of S is contained in ℤ. Since this is a statement about cohomology, Weinstein raises the question of giving a purely algebraic topology proof of this result.
R. Lashof

Momentum Mappings And Reduction of Poisson Actions

Abstract
An action σ: G × PP of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : PP for gG preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map PG \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: Pg*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µg* and G µ G is the coadjoint isotropy subgroup of µ.
Jiang-Hua Lu

On Jacobi Manifolds and Jacobi Bundles

Abstract
We introduce the notion of a Jacobi bundle, which generalizes that of a Jacobi manifold. The construction of a Jacobi bundle over a conformal Jacobi manifold has, as particular cases, the constructions made by A. Weinstein [21] of a Le Brun-Poisson manifold over a contact manifold, and that of a Heisenberg-Poisson manifold over a symplectic (or Poisson) manifold. We show that the total space of a Jacobi bundle has a natural homogeneous Poisson structure, and that with each section of that bundle is associated a Hamiltonian vector field, defined on the total space of the bundle, tangent to the zero section, which projects onto the base manifold.
Charles-Michel Marle

Groupes de Lie à Structure Symplectique Invariante

Résumé
Un groupe de Lie G admet une structure symplectique invariante s’il existe sur G une 2-forme différentielle fermée invariante à gauche dont le rang est égal à la dimension de G. Un tel groupe sera appelé par abus de langage, symplectique et son algèbre de Lie sera dite symplectique. Le principal résultat de ce travail est de fournir une classification des groupes symplectiques nilpotents par leurs algèbres de Lie. L’idée centrale dans cette classification est la notion de double extension (section 2) d’une algèbre symplectique: grosso modo en additionnant un plan symplectique à une algèbre symplectique on obtient une algèbre symplectique. Cette notion est l’analogue symplectique de la notion de double extension des algèbres de Lie orthogonales, que nous avons introduite dans [Me-Re 1].Nous montrons que toute algèbre symplectique nilpotente s’obtient par une suite de doubles extensions à partir de l’algèbre réduite à zéro (théorème 2.5). Dire que le groupe de Lie symplectique (G,ω) est double extension du groupe (H, Ω) veut dire que ce dernier est une variété réduite de Marsden-Weinstein de (G,ω). Toute nilvariété symplectique étant quotient d’un groupe nilpotent symplectique par un sous-groupe discret co-compact, [Be-Go] la double extension permet d’obtenir toutes ces variétés.
Alberto Medina, Philippe Revoy

Holonomy Groupoids of Generalized Foliations. II. Transverse Measures and Modular Classes

Abstract
A generalized foliation is a foliation with singular leaves in the sense of P. Stefan [St] and P. Dazord [D]. In the preceding paper [Su], we defined notions of holonomy maps and holonomy groupoids for a generalized foliation whose singular leaves are all tractable. We continue to investigate their properties.
Haruo Suzuki

Symplectic Groupoids, Geometric Quantization, and Irrational Rotation Algebras

Abstract
The rotation algebra A θ for a real parameter θ is defined as the crossed product of the additive group ℤ with the space of functions on the circle T = ℝ/2πℤ via the action n · φ = φ+nθ. 1 More concretely, A θ is the completion with respect to a certain norm of its subalgebra A θ of smooth elements. The elements of A θ are the smooth functions f (n, φ) on ℤ × T which, with all their φ-derivatives, are rapidly decreasing in n. The multiplication, which we denote by *, is defined by
$$\left( {f * g} \right)\left( {n,\varphi } \right) = \sum\limits_{m,k \in z,m + k = n} {f\left( {m,\varphi } \right)} g\left( {k,\varphi + m\theta } \right)$$
Alan Weinstein

Morita Equivalent Symplectic Groupoids

Abstract
Morita equivalence of C*-algebras, first introduced by M. Rieffel, has been widely accepted as one of the most important equivalence relations in C*-algebras [Riel] [Rie2] [Rie3] [Rie4]. Roughly speaking, two C*- algebras are said to be Morita equivalent if there is an equivalence bimodule between them. Morita equivalent C*-algebras have many similar features. For instance, they have equivalent categories of left modules, isomorphic K-groups, and so on. Also, Morita equivalence plays a very important role in understanding the structure of some C*-algebras such as transformation C*-algebras and foliation C*-algebras. A natural question arises as to what the classical analogue of this equivalence relation is. It is generally accepted that the classical analogue of a C*-algebra (or non-commutative algebra) is a Poisson manifold. So, more precisely, we expect to find an equivalence relation for Poisson manifolds that plays the same role as Morita equivalence does for C*-algebras. A solution to this problem was made possible by the recent introduction of symplectic groupoids in the study of Poisson manifolds due to Karasev and Weinstein [CDW] [Ka] [W2]. The original purpose for introducing symplectic groupoids was to study nonlinear commutation relations and quantization theory. In fact, it turns out that symplectic groupoids provide a bridge between Poisson manifolds, C*-algebras, as well as quantizations. Therefore, introducing and studying Morita equivalence of symplectic groupoids should be the first step in understanding Morita equivalence of Poisson manifolds.
Ping Xu
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