On the role of effective representations of Lie groupoids

Details On the role of effective representations of Lie groupoids On the role of effective representations of Lie groupoids

2009-03-31

arxiv.org/abs/0903.5422v1

Advances in Mathematics 225 (2) (2010), pp. 826-858

Mathematics

Representation Theory

32 pages

Scientific paper

10.1016/j.aim.2010.03.014

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.

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