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 concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus pr...

The Lagrange-d’Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d’Alembert-Poincaré equations. In a previous contribution we have shown that both sets of equations fall in the category of so-called ‘Lagrangian systems on a subbundle of a Lie algebroid’. In this paper, we investigate the special case when the reduced system is again invariant under a new ...

<abstract><p>In this paper, first we introduce the notion of a $ \mathsf{VB} $-Lie 2 $-algebroid, which can be viewed as categorification algebroid. The tangent prolongation Lie $-algebroid is naturally. We show that after choosing splitting, there one-to-one correspondence between $-algebroids and flat superconnections 2-algebroid on 3-term complex vector bundles. Then $-$ \mathsf{...

We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle QA = ∧topA⊗ ∧topT ∗P . The line bundle QA may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of QA may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H(A) called the modular c...

A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of d...

Axioms of Lie algebroid are discussed. In particular, it is shown that a Lie quasi-algebroid (i.e. a Lie algebra bracket on the C∞(M)-module E of sections of a vector bundle E over a manifold M which satisfies [X, fY ] = f [X, Y ] + A(X,f)Y for all X, Y ∈ E , f ∈ C∞(M), and for certain A(X,f) ∈ C∞(M)) is a Lie algebroid if rank(E) > 1, and is a local Lie algebra in the sense of Kirillov if E is...

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describ...

We study holomorphic Poisson manifolds, holomorphic Lie algebroids and holomorphic Lie groupoids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bu...

We introduce an n-ary Lie algebroid canonically associated to a NambuPoisson manifold. We also prove that every Nambu-Poisson bracket defined on functions is induced by some differential operator on the exterior algebra, and characterize such operators. Some physical examples are presented.