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Pré-Publication, Document De Travail Année : 2015

Invariant connections and PBW theorem for Lie groupoid pairs

Résumé

Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we interpret the associated Atiyah class 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 Poincaré–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 simultaneous linearization of all the monodromies. In the course of the paper, a general theory of connections and connection forms on Lie groupoid principal bundles is developed. Also, a computational substitute to the adjoint action (which only exists " up to homotopy ") is suggested .
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Dates et versions

hal-01285656 , version 1 (09-03-2016)

Identifiants

  • HAL Id : hal-01285656 , version 1

Citer

Camille Laurent-Gengoux, Yannick Voglaire. Invariant connections and PBW theorem for Lie groupoid pairs. 2015. ⟨hal-01285656⟩

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