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Invariant connections and PBW theorem fo...
Laurent-Gengoux, Cam...

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Invariant connections and PBW theorem for Lie groupoid pairs by Laurent-Gengoux, Camille ( Author )
Lemonjar Open Access
N.A
04-07-2015
To a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with vanishing Atiyah class, we show that the left A-action on the quotient space L/A can be linearized. In addition to giving an alternative proof of a result of Calaque about the Poincare-Birkhoff-Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to the simultaneous linearization of all the monodromies. In the course of the paper, a general theory of connections on Lie groupoid equivariant principal bundles is developed.
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https://arxiv.org/abs/1507.01051
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