We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.

We recall the concept of a Lie algebroid on a vector bundle and the associated notion of Lagrange-type equations. A heuristic calculus of variations approach tells us what a time-dependent generalization of such equations should look like. In order to find a geometrical model for such a generalization, the idea of a Lie algebroid structure on a class of affine bundles is introduced. We develop ...

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H , and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra as...

We propose a new framework for constructing geometric and physical models on spacetimes provided with Lie algebroid symmetry, i.e. manifolds provided with additional anchor and generalized Lie algebra commutator structures. The approach is related to the geometry of moving nonholonomic frames with associated nonlinear connections. A strict application of such geometric methods to spinor fields ...

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid 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.

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras 1 , in the theory of integrable system 2 , and in field theory 3 . But for instance, in 2 , the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In 4, 5 we have defined such a test functional space in the case of a linear Poisson...

We consider the category of anchored Banach vector bundles and we discuss the notion of semispray. Adding on the set of sections of an anchored Banach vector bundle a Lie bracket with some properties one gets the notion of Lie algebroid. We prove that the Lie algebroids form also a category. A Dirac structure on a Banach manifold M is defined as a subbundle of the big tangent bundle TM ⊕ T ∗M t...

To any manifold equipped with a higher degree closed form, one can associate an L∞-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this L∞-algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produ...

We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid G in terms of the cohomology of S * (G), the cosphere bundle of A(G), where A(G) is the Lie algebroid of G. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space A * (G) 0 ∼ = S * (G) × (0, ∞), the dual of A(G) with the zero section removed. ...