Given a Poisson manifold P, if there exists a symplectic manifold Σ and a surjective submersion Σ → P then it is possible to quantize Σ and then “push” the results to P. This method of quantizing a Poisson manifold is known as symplectic realisation. In this paper we illustrate how this method is related with the integrability of Lie algebroids.

Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.