Rigidity of Poisson Lie group actions

n this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum maps (when the acting Lie group G is endowed with an arbitrary Poisson structure) combining a rigidity result for canonical Hamiltonian actions (\cite{MMZ}) and a linearization theorem(\cite{GW}). These results have applications to quantization of symmetries since these infinitesimal momentum maps appear as the classical limit of quantum momentum maps (\cite{BEN}).Peer ReviewedPreprin

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.Peer ReviewedPostprint (updated version

A normal form theorem for integrable systems on contact manifolds

We present a normal form theorem for singular integrable systems on contact manifoldsPostprint (published version

The geometry of E-manifolds

Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an E-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting generalizes the study of b-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser's theorem to this setting, and use it to construct symplectomorphisms between singular symplectic forms. We give appli- cations of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in depth.Preprin

Integrable systems and group actions

The main purpose of this paper is to present in a uni¯ed approach to see di®erent results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.Preprin

From action-angle coordinates to geometric quantization

The philosophy of geometric quantization is to ¯nd and understand a \(one-way) dictionary" that \translates" classical systems into quantum systems . In this way, a quantum system is associated to a classical system in which observables (smooth functions) become operators of a Hilbert space and the classical Poisson bracket becomes the commutator of operators. In this process, the choice of additional geometric structures (polarizations) plays an important r^ole. A desired property is that the quantization obtained does not depend on the polarization. Another rule in the game is that of keeping track of the symmetries on both sides. This is the deep link of geometric quantization with representation theory. The quanti- zation commutes with reduction \principle" becomes realistic in some geometric quantization set-ups. Our point of view in this big endeavour is very modest. We plan to construct a \representation space" in the case the polarization is given by a real polarization. For this, we follow the de¯nition of Kostant of the representation spaces via higher cohomology groups with coe±cients in the sheaf of °at sections. In this short note, we will not discuss either the (pre)Hilbert structure of this space nor the quantization rules.Preprin

Desingularizing b^m-symplectic structures

A 2n-dimensional Poisson manifold (M,¿) is said to be bm-symplectic if it is symplectic on the complement of a hypersurface Z and has a simple Darboux canonical form at points of Z which we will describe below. In this paper we will discuss a desingularization procedure which, for m even, converts ¿ into a family of symplectic forms ¿¿ having the property that ¿¿ is equal to the bm-symplectic form dual to ¿ outside an ¿-neighborhood of Z and, in addition, converges to this form as ¿ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of bm-manifolds can be more clearly understood by viewing them as limits of analogous properties of the ¿¿'s. We will also prove versions of these results for m odd; however, in the odd case the family ¿¿ has to be replaced by a family of folded symplectic forms.Preprin

The clay public lecture and conference on the Poincaré Conjecture, Paris, 7-9 June 2010

Postprint (published version

Symplectic and poisson structures with symmetries in interaction

Hamiltonian actions constitute a central object of study in symplectic geometry. Special attention has been devoted to the toric case. Toric symplectic manifolds provide natural examples of integrable systems and every integrable system on a symplectic manifold is a toric manifold in a neighbourhood of a compact fiber (Arnold–Liouville). The classification of toric symplectic manifolds is given by Delzant’s theorem in terms of the image of the moment map (Delzant polytope)....Postprint (published version

Cotangent models for integrable systems on bb-symplectic manifolds

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