Marius Crainic

Integration of twisted Dirac brackets

Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi on M correspond to maps from the Lie algebroid of G into T* M satisfying an algebraic condition and a differential condition with respect to the phi-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to phi-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.

Representations up to homotopy of Lie algebroids

Abstract We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkmans BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [3].

Cyclic cohomology of étale groupoids; The general case

We give a general method for computing the cyclic cohomology of crossed products by etale groupoids extending the FeiginTsyganNistor spectral sequences In particular we extend the computations performed by Brylinski Burghelea Connes Feigin Karoubi Nistor and Tsygan for the convolution algebra C c G of an etale groupoid removing the Hausdorness condition and including the computation of hyperbolic components Examples like group actions on manifolds and foliations are considered

Dirac structures, momentum maps, and quasi-Poisson manifolds

We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and Hamiltonian actions, to the realm of Dirac geometry. As an example, we show how Hamiltonian quasi-Poisson manifolds fit into this framework by constructing an “inversion” procedure relating quasi-Poisson bivectors to twisted Dirac structures.

Deformations of Lie Groupoids

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Mosers deformation arguments for groupoids, we obtain several rigidity and normal form results.

Poisson Manifolds of Compact Types (PMCT 1)

Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

Orbispaces as differentiable stratified spaces

We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures—it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.