Foliated Lie and Courant Algebroids

  title={Foliated Lie and Courant Algebroids},
  author={Izu Vaisman},
  journal={Mediterranean Journal of Mathematics},
  • I. Vaisman
  • Published 8 February 2009
  • Mathematics
  • Mediterranean Journal of Mathematics
If A is a Lie algebroid over a foliated manifold $${(M, {\mathcal {F}})}$$, a foliation of A is a Lie subalgebroid B with anchor image $${T{\mathcal {F}}}$$ and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of $${\mathcal F}$$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's supervector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In… 
LA-Courant Algebroids and their Applications
In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d,
Almost Kähler Ricci Flows and Einstein and Lagrange–Finsler Structures on Lie Algebroids
In this work we investigate Ricci flows of almost Kähler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or (effective)
M. JOTZ AND C. ORTIZAbstract. In this work, we study Lie groupoids equipped with multiplicative foliationsand the corresponding infinitesimal data. We determine the infinitesimal counterpartof a
Foliated groupoids and infinitesimal ideal systems
A construction of Courant algebroids on foliated manifolds
For any transversal-Courant algebroid E on a foliated manifold (M,F), and for any choice of a decomposition T M = TF © Q, we construct a
Poisson structures on almost complex Lie algebroids
In this paper, we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex
On the geometry of double field theory
Double field theory was developed by theoretical physicists as a way to encompass T-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms in the
On Almost Complex Lie Algebroids
The almost complex Lie algebroids over smooth manifolds are considered in the paper. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost


Manin Triples for Lie Bialgebroids
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does
Transitive Courant algebroids
A class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection is described.
Lie bialgebroids and Poisson groupoids
Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a
Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson
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 (Theorem 4.1 in Crainic,
Lie Algebroids, Holonomy and Characteristic Classes
Abstract We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual
Reduction of branes in generalized complex geometry
We show that certain submanifolds of generalized complex manifolds (“weak branes”) admit a natural quotient which inherits a generalized complex structure. This is analog to quotienting coisotropic
We define integrable, big-isotropic structures on a manifold M as subbundles E ⊆ TM ⊕ T*M that are isotropic with respect to the natural, neutral metric (pairing) g of TM ⊕ T*M and are closed by