Room P3.10, Mathematics Building

Giorgio Trentinaglia, Goettingen, Germany
On the role of effective representations of proper Lie groupoids

In this talk, we discuss the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid $G$, the canonical homomorphism of $G$ into the reconstructed groupoid $T (G)$ is surjective, although -contrary to what happens in the case of groups- it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that $G$ may be isomorphic to $T (G)$ and, more generally, in order that $T (G)$ may be a Lie groupoid. We show that if $T (G)$ is a Lie groupoid, the canonical homomorphism $G \to T(G)$ is a submersion and the two groupoids have isomorphic categories of representations.