In this paper the properties of Lie algebroids with Poisson structures are investigated. We generalize some results of Fernandes [1] regarding linear contravariant connections on Poisson manifolds at the level of Lie algebroids. In the last part, the notions of complete and horizontal lifts on the prolongation of Lie algebroid are studied and their compatibility conditions are pointed out.

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

Consider a complex symplectic manifold X and the algebroid stack WX of deformation-quantization. For two regular holonomic WXmodules Li (i = 0, 1) supported by smooth Lagrangian submanifolds, we prove that the complex RHom WX (L1,L0) is perverse over the field Wpt and dual to the complex RHomWX(L0,L1).

We give a new proof of the equivalence between existence complete action Lie algebroid on surjective submersion and its integrability. The main tools in our approach are double groupoids multiplicative foliations, relies simple characterization those vacant which induce groupoid structure their orbit spaces.

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal ,4-modules is studied in the special case in which A is the ring of integers in the field obtained by adjoining pth roots of unity to Qp , the p-adic numbers. Information about these cohomology groups is used to give new proofs of results about the E2 term of the Adams spectral sequence based on 2-local...

In this paper, we construct a homotopy Poisson algebra of degree 3 associated to split Lie 2-algebroid, by which give new approach characterize 2-bialgebroid. We develop the differential calculus 2-algebroid and establish Manin triple theory for 2-algebroids. More precisely, notion strict Dirac structure define 2-algebroids be CLWX with two transversal structures. show that there is one-to-one ...

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundn...