Reference : Invariant connections and PBW theorem for Lie groupoid pairs
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Physical, chemical, mathematical & earth Sciences : Mathematics
Invariant connections and PBW theorem for Lie groupoid pairs
Laurent-Gengoux, Camille []
Voglaire, Yannick mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] Given a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid pairs with vanishing Atiyah class, we show that the left
$\mathbf{A}$-action on the quotient space $\mathbf{L}/\mathbf{A}$ 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 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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