Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

By M. Crainic. 2000

255 citations recorded by the source corpus.

AbstractIn the first section we discuss Morita invariance of differentiable/algebroid cohomology.In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14].In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the Van Est map. This extends Evens-Lu-Weinstein’s characteristic class $\theta_{L}$ [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 30].In the last section we describe applications to Poisson geometry.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/03101e1d953763ca0453eeab77cbb5701416cd0a

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In Algebra and Number Theory

1. M. Crainic
2. R. Mehta
3. C. A. Abad
4. A. Gracia-Saz
5. H. Bursztyn
6. I. Struchiner
7. P. Xu
8. K. Behrend

In Mathematical Physics

1. M. Crainic
2. R. Mehta
3. C. A. Abad
4. H. Bursztyn
5. A. Gracia-Saz
6. I. Struchiner
7. P. Xu
8. K. Behrend

Citations in the corpus

• On the linearization theorem for proper Lie groupoids (2011, background): “Recall that any proper Lie groupoid H over a manifold N admits a family of linear maps (constructed from a Haar system and a cut-off function [3]) ∫”
• Q-algebroids and their cohomology (2007, background): “In [28], the relationship is explored further by showing that the van Est homomorphism [6, 39] extends to a morphism of double complexes.”
• The Weil algebra and the Van Est isomorphism (2009, methodology): “where the last equation is a direct application of Theorem 2 from [ 7 ].”
• Supergroupoids, double structures, and equivariant cohomology (2006, methodology): “In Chapter 6, we describe an extension of the van Est map [ 16 ] [72] [76] that maps”
• Poisson geometry and Morita equivalence (2004, background): “Morita equivalence preserves first Poisson cohomology groups [28, 40], and modular classes [28, 91];”
• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, methodology): “We give a complete classification of regular VB–algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.”
• Lectures on integrability of Lie brackets (2006, unknown): “M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom.”
• Representations up to homotopy and Bott's spectral sequence for Lie groupoids (2009, background): “The vanishing for the pair groupoid can be seen as a consequence of another important property of differentiable cohomology: its Morita invariance (see [7]).”
• VB-groupoids and representation theory of Lie groupoids (2010, methodology): “It is known [2, 3, 6, 7, 11] that the modular class is only the first of a sequence of “secondary” characteristic classes associated to Lie algebroid representations up to homotopy.”