We review the extent to which the structure of the universal enveloping algebra of a Lie algebroid over a manifold resembles a Hopf algebra, and prove a Cartier-Milnor-Moore theorem for this type of structure.
COBISS.SI-ID: 15636569
In this paper we give an extension of the Cartier-Gabriel-Kostant structure theorem to Hopf algebroids.
COBISS.SI-ID: 16432473
The classical Serre-Swan's theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of etale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.
COBISS.SI-ID: 16096857
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of the homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
COBISS.SI-ID: 16434009
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to presentability of proper regular Lie groupoids.
COBISS.SI-ID: 16400729