Connections up to homotopy and characteristic classes

In this note we clarify the relevance of ``connections up to homotopy'' to the theory of characteristic classes. We have already remarked \cite{Crai} that such connections up to homotopy can be used to compute the classical Chern characters. Here we present a slightly different argument for this, and then proceed with the discussion of the flat (secondary) characteristic classes. As an application, we clarify the relation between the two different approaches to characteristic classes of algebroids (and of Poisson manifolds in particular): we explain that the intrinsic characteristic classes are precisely the secondary classes of the adjoint representation.Comment: 12 page

Chern characters via connections up to homotopy

The aim of this note is to point out that Chern characters can be computed using curvatures o ``super-connections up to homotopy'. We also present an application to the vanishing theorem for Lie algebroids which is at the origin of new secondary classes of algebroids (Fernandes), hence, in particular, of Poisson manifolds.Comment: 6 page

Deformations of Lie brackets and representations up to homotopy

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.Comment: 28 page

Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory

REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern-Weil theory.Comment: Completely revised version (new results added); 38 page

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

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. 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). As a second application we extend van Est's argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives (a slight improvement of) Hector-Dazord's integrability criterion. 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 $\theta_{L}$ (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles. In the last section we describe some applications to Poisson geometry (e.g. we clarify the Morita invariance of Poisson cohomology, and of the modular class).Comment: 37 page

Deformations of Lie brackets: cohomological aspects

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which ``controls'' deformations of the structure bracket of the algebroid. We also have a closer look at various special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra actions on manifolds.Comment: 17 pages, Revised version: small corrections, more references adde