Invariant connections and PBW theorem for Lie groupoid pairs

TL;DR

This paper studies the structure of Lie groupoid pairs, introduces an Atiyah class as an obstruction to invariant connections, and explores conditions for linearization and PBW theorems, providing new insights into their geometric and algebraic properties.

Contribution

It develops a theory of invariant connections on Lie groupoid pairs, relates the Atiyah class to linearization conditions, and offers an alternative proof of PBW theorem for Lie algebroid pairs.

Findings

Atiyah class as an obstruction to invariant connections

Linearization criteria for Lie groupoid pairs with vanishing Atiyah class

Interpretation of Molino class as an obstruction to monodromy linearization

Abstract

To a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret 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 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.