Some results on the Mackenzie obstruction for transitive Lie algebroids

Differential Operators and Actions of Lie Algebroids

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...

On Poisson Realizations of Transitive Lie Algebroids

We show that every transitive Lie algebroid over a connected symplectic manifold comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure. The reconstruction of the corresponding Poisson structures from the Lie algebroid is given in terms of coupling tensors.

Horizontal Subbundle on Lie Algebroids

Providing an appropriate definition of a horizontal subbundle of a Lie algebroid will lead to construction of a better framework on Lie algebriods. In this paper, we give a new and natural definition of a horizontal subbundle using the prolongation of a Lie algebroid and then we show that any linear connection on a Lie algebroid generates a horizontal subbundle and vice versa. The same correspo...

Notions of Double for Lie Algebroids

Drinfel’d Doubles and Ehresmann Doubles for Lie Algebroids and Lie Bialgebroids

We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (w...