A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but i...

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed non-degenerate n + 1-form. The case relevant to classical string theory is when n = 2 and is called ‘2-plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2-...

Consider a ring K, a topological space X and a sheaf A on X of K[[~]]-algebras. Assuming A ~-adically complete and without ~torsion, we first show how to deduce a coherency theorem for complexes of A -modules from the corresponding property for complexes of A /~A -modules. We apply this result to prove that, under a natural properness condition, the convolution of two coherent kernels over defo...

In this paper we set up the foundations around notions of formal differentiation and integration in context commutative Hopf algebroids Lie-Rinehart algebras. Specifically, construct a contravariant functor from category with fixed base algebra to that algebras over same algebra, functor, which can be seen as an algebraic counterpart process Lie groupoids algebroids. The other way around, provi...

Several new invariants of Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.

In this note we determine the obstruction to triviality of the stack of exact vertex algebroids thereby recovering the result of [GMS]. The stack EVAOX of exact vertex OX-algebroids is a torsor under the stack in Picard groupoids ECAOX of exact Courant OX -algebroids. The latter is equivalent to the stack of torsors under ΩX −→Ω . Therefore, ECAOX -torsors are classified by H (X; ΩX −→Ω ). The ...

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point.