Abstract

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.

We also develop a general theory of connections on Lie groupoid equivariant principal bundles.

Keywords

Mathematical Subject Classification 2010

Milestones

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