We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequence. We also give a description of the groupoid integrating an extension in ...

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 paper, we extend the T-duality isomorphism by Gualtieri and Cavalcanti, from invariant exact Courant algebroids, to exotic algebroids such that momentum winding numbers are exchanged, filling in a gap literature.

We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representation of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how these notions extend to derivative representations of Lie algebroids and semi-linear representations of Lie g...

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of com...

Consider a ring K, a topological space X and a sheaf A on X of K[[~]]-algebras. Assuming that A is ~-complete and without ~torsion, we first show how to deduce a coherency theorem for complexes of A -modules from a 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 deforma...

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and ...