90 research outputs found
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
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
.
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
(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
Dirac structures, moment 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.Comment: 36 pages. Typos and signs fixed. To appear in Progress in
Mathematics, Festschrift in honor of Alan Weinstein, Birkause
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