467 research outputs found
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
Foliation groupoids and their cyclic homology
In this paper we study the Lie groupoids which appear in foliation theory. A
foliation groupoid is a Lie groupoid which integrates a foliation, or,
equivalently, whose anchor map is injective. The first theorem shows that, for
a Lie groupoid G, the following are equivalent:
- G is a foliation groupoid,
- G has discrete isotropy groups,
- G is Morita equivalent to an etale groupoid.
Moreover, we show that among the Lie groupoids integrating a given foliation,
the holonomy and the monodromy groupoids are extreme examples.
The second theorem shows that the cyclic homology of convolution algebras of
foliation groupoids is invariant under Morita equivalence of groupoids, and we
give explicit formulas. Combined with the previous results of Brylinski, Nistor
and the authors, this theorem completes the computation of cyclic homology for
various foliation groupoids, like the (full) holonomy/monodromy groupoid, Lie
groupoids modeling orbifolds, and crossed products by actions of Lie groups
with finite stabilizers. Some parts of the proof, such as the H-unitality of
convolution algebras, apply to general Lie groupoids.
Since one of our motivation is a better understanding of various approaches
to longitudinal index theorems for foliations, we have added a few brief
comments at the end of the second section.Comment: 18 page
Cyclic cohomology of Hopf algebras
We give a construction of ConnesMoscovicis cyclic cohomology for any Hopf algebra equipped with a character Furthermore we introduce a noncommutative 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 noncommutative version of the usual ChernWeil theor
Integration of Dirac-Jacobi structures
We study precontact groupoids whose infinitesimal counterparts are
Dirac-Jacobi structures. These geometric objects generalize contact groupoids.
We also explain the relationship between precontact groupoids and homogeneous
presymplectic groupoids. Finally, we present some examples of precontact
groupoids.Comment: 10 pages. Brief changes in the introduction. References update
Integration of twisted Dirac brackets
The correspondence between Poisson structures and symplectic groupoids,
analogous to the one of Lie algebras and Lie groups, plays an important role in
Poisson geometry; it offers, in particular, a unifying framework for the study
of hamiltonian and Poisson actions. In this paper, we extend this
correspondence to the context of Dirac structures twisted by a closed 3-form.
More generally, given a Lie groupoid over a manifold , we show that
multiplicative 2-forms on relatively closed with respect to a closed 3-form
on correspond to maps from the Lie algebroid of into the
cotangent bundle of , satisfying an algebraic condition and a
differential condition with respect to the -twisted Courant bracket. This
correspondence describes, as a special case, the global objects associated to
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.Comment: 42 pages. Minor changes, typos corrected. Revised version to appear
in Duke Math.
Integration of Holomorphic Lie Algebroids
We prove that a holomorphic Lie algebroid is integrable if, and only if, its
underlying real Lie algebroid is integrable. Thus the integrability criteria of
Crainic-Fernandes do also apply in the holomorphic context without any
modification. As a consequence we give another proof of the following theorem:
a holomorphic Poisson manifold is integrable if, and only if, its real (or
imaginary) part is integrable as a real Poisson manifold.Comment: 26 pages, second part of arXiv:0707.4253 which was split into two,
v2: example 3.19 and section 3.7 adde
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