Edith Padrón

Leibniz algebroid associated with a Nambu-Poisson structure

The notion of a Leibniz algebroid is introduced, and it is shown that each Nambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact permits one to define the modular class of a Nambu-Poisson manifold as an appropriate cohomology class, extending the well known modular class of Poisson manifolds.

On the computation of the Lichnerowicz–Jacobi cohomology

Abstract Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples.

On the geometric quantization of Jacobi manifolds

The geometric quantization of Jacobi manifolds is discussed. A natural cohomology (termed Lichnerowicz–Jacobi) on a Jacobi manifold is introduced, and using it the existence of prequantization bundles is characterized. To do this, a notion of contravariant derivatives is used, in such a way that the procedure developed by Vaisman for Poisson manifolds is naturally extended. A notion of polarization is discussed and the quantization problem is studied. The existence of prequantization representations is also considered.

Lagrangian submanifolds and dynamics on lie affgebroids

In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange-Poincaré (Hamilton-Poincaré) equations are the Euler-Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that LagrangePoincaré (Hamilton-Poincaré) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids. Mathematics Subject Classification (2000): 17B66, 53D12, 70G45, 70H03, 70H05, 70H20.

Jacobi-Nijenhuis manifolds and compatible Jacobi structures

Abstract We propose a definition of Jacobi—Nijenhuis structures, that includes the Poisson—Nijenhuis structures as a particular case. The existence of a hierarchy of compatible Jacobi structures on a Jacobi—Nijenhuis manifold is also obtained.

Lichnerowicz-Jacobi cohomology

In this paper we extend the notion of Lichnerowicz - Poisson cohomology to Jacobi manifolds. We study the relation of the so-called Lichnerowicz - Jacobi cohomology with the basic de Rham cohomology and the cohomology of the Lie algebra of functions relative to the representation defined by the Hamiltonian vector fields. A natural pairing with the canonical homology is constructed. The relation between the Lichnerowicz - Poisson cohomology of a quantizable Poisson manifold and the Lichnerowicz - Jacobi cohomology of the total space of a prequantization bundle is obtained. Particular cases of cosymplectic, contact and locally conformal symplectic manifolds are discussed. Finally, the Lichnerowicz - Jacobi cohomology of a non-transitive example is studied.

A unified framework for mechanics: Hamilton–Jacobi equation and applications

In this paper, we construct Hamilton-Jacobi equations for a great variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems...). We recover all these, in principle, different cases using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory.

La cohomologie de Lichnerowicz-Jacobi pour les variétés de Jacobi

Abstract A cohomology for Jacobi manifolds which generalizes the Lichnerowicz—Poisson cohomology for Poisson manifolds is constructed.

Duality and modular class of a Nambu-Poisson structure

In this paper we introduce cohomology and homology theories for Nambu-Poisson manifolds. Also we study the relation between the existence of a duality for these theories and the vanishing of a particular Nambu-Poisson cohomology class, the modular class. The case of a regular Nambu-Poisson structure and some singular examples are discussed.

Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group

We describe the reduction procedure for a symplectic Lie algebroid by a Lie sub- algebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.