On the role of effective representations of Lie groupoids

TL;DR

This paper explores the Tannaka duality for representations of proper Lie groupoids, establishing conditions under which the original groupoid and its reconstruction are isomorphic or share representation categories.

Contribution

It provides necessary and sufficient conditions for a proper Lie groupoid to be isomorphic to its Tannaka reconstruction and analyzes when the reconstructed groupoid is a Lie groupoid.

Findings

The canonical homomorphism from G to T(G) is surjective for proper Lie groupoids.

G may not be isomorphic to T(G), unlike the case for groups.

If T(G) is a Lie groupoid, the homomorphism is a submersion and the categories of representations are equivalent.

Abstract

In this paper, we undertake the study of 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 from G into T(G) is a submersion and the two groupoids have isomorphic categories of representations.