We propose a generalization of the momentum map on symplectic manifold with Lie algebra action to Courant algebroid structure. The theory section is generalized compatible algebroid. As an example, we identify in constrained Hamiltonian mechanics symmetry. Moreover, construct cohomological formulations by considering BFV and BV formalism this system. Weil for structure constructed.

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17...

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a E1(M)-Dirac structure L induces on every leaf F of its characteristic foliation a E1(F )-Dirac structure LF , which comes from a precontact structure or from a locally conformal pr...

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie alge...

In this paper we discuss the relation between the unimodularity of a Lie algebroid τA : A → Q and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle A. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion subalgebroid by dropping constant rank requirement. lay bases a theory for subalgebroids: we construct associated holonomy groupoids, adapting procedure Androulidakis-Skandalis foliations, in way that keeps track choice groupoid integrating ambient algebroid. The groupoids are topological and suitable nonco...

Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of morphisms of Lie n-algebroids over different bases, give an explicit formula fo...

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions...

In this paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found in the more general case of Lie algebroids. We prove that the canonical nonlinear connection induced by a regular Lagrangian on a Lie algebroid is the unique co...