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13
Integrability of Poisson brackets
, 2004
"... We discuss the integration of Poisson brackets, motivated by our recent solution to the integrability problem for general Lie brackets. We give the precise obstructions to integrating Poisson manifolds, describing the integration as a symplectic quotient, in the spirit of the Poisson sigmamodel of ..."
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Cited by 39 (9 self)
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We discuss the integration of Poisson brackets, motivated by our recent solution to the integrability problem for general Lie brackets. We give the precise obstructions to integrating Poisson manifolds, describing the integration as a symplectic quotient, in the spirit of the Poisson sigmamodel of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas, improving on results of Alcalde Cuesta and Hector. We apply our results (and our point of view) to decide about the existence of complete symplectic realizations, to the integrability of submanifolds of Poisson manifolds, and to the study of dual pairs, Morita equivalence and reduction.
Lie algebroids, holonomy and characteristic classes
 Adv. Math
"... Abstract. We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit folia ..."
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Cited by 38 (6 self)
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Abstract. We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit foliation of a Lie algebroid and prove a stability theorem. We also introduce secondary or exotic characteristic classes that generalize the modular class of a Lie algebroid. Introduction and Basic Definitions The theory of connections is a classical topic in differential geometry. They provide an extremely important tool to study geometric structures on manifolds and, as such, they have been applied with great success in many different settings. However, the use of connections has been very limited whenever singular behavior
L.: Connections in Poisson geometry. I. Holonomy and invariants
 J. Differential Geom
"... Abstract. We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by I. Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds ..."
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Cited by 27 (5 self)
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Abstract. We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by I. Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.
Coupling tensors and Poisson geometry near a single symplectic leaf. In: Lie algebroids and related topics in differential geometry
 Banach Center Publ., Vol
, 2001
"... In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on a fiber bundle, called coupling tensors. We show that every Poisson structure near a closed symplectic leaf can be realized as a co ..."
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Cited by 18 (0 self)
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In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on a fiber bundle, called coupling tensors. We show that every Poisson structure near a closed symplectic leaf can be realized as a coupling tensor. Our main result is a geometric criterion for the neighborhood equivalence between Poisson structures with the same leaf. This criterion gives a Poisson version of the relative Darboux theorem due to Weinstein. Within the category of the algebroids, coupling tensors are introduced on the dual of the isotropy of a transitive Lie algebroid over a symplectic base. As a basic application of these results, we show that there is a well defined notion of a “linearized ” Poisson structure over a closed symplectic leaf which gives rise to a natural model for the linearization problem. The paper is based on a lecture delivered at the International Conference
Grothendieck Groups of Poisson Vector Bundles
"... Abstract. A new invariant of Poisson manifolds, a Poisson Kring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson Kring are proved and ..."
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Cited by 16 (0 self)
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Abstract. A new invariant of Poisson manifolds, a Poisson Kring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson Kring are proved and the Poisson Krings are calculated for a number of examples. In particular, for the zero Poisson structure the Kring is the ordinary K 0ring of the manifold and for the dual space to a Lie algebra the Kring is the ring of virtual representations of the Lie algebra. It is also shown that the Kring is an invariant of Morita equivalence. Moreover, the Kring is a functor on a category, the weak
Nonlinearizability of certain Poisson structures near a symplectic leaf
"... We give an intrinsic proof that Vorobjev’s first approximation of a Poisson manifold near a symplectic leaf is a Poisson manifold. We also show that Conn’s linearization results cannot be extended in Vorobjev’s setting. 1 ..."
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Cited by 1 (0 self)
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We give an intrinsic proof that Vorobjev’s first approximation of a Poisson manifold near a symplectic leaf is a Poisson manifold. We also show that Conn’s linearization results cannot be extended in Vorobjev’s setting. 1
Poisson Equivalence over a Symplectic Leaf
, 2005
"... We study the equivalence of Poisson structures around a given symplectic leaf of nonzero dimension. Some criteria of Poisson equivalence are derived from a homotopy argument for coupling Poisson structures. In the case when the transverse Lie algebra of the symplectic leaf is semisimple of compact t ..."
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Cited by 1 (0 self)
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We study the equivalence of Poisson structures around a given symplectic leaf of nonzero dimension. Some criteria of Poisson equivalence are derived from a homotopy argument for coupling Poisson structures. In the case when the transverse Lie algebra of the symplectic leaf is semisimple of compact type, we show that an obstruction to the linearizability is the cohomology class of a Casimir 2cocycle. This allows us to obtain a semilocal analog of the Conn linearization theorem and to clarify examples of nonlinearizable Poisson structures due to [DW]. 1
Invariants of Lie algebroids
 Differential Geom. Appl
"... Abstract. Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and we established several relationships between them. 1. ..."
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Cited by 1 (0 self)
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Abstract. Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and we established several relationships between them. 1.