Invariant connections and PBW theorem for Lie groupoid pairs

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.

49 pages, final version accepted for publication in the Pacific Journal of Mathematics

References in corpus (6)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Lie-Rinehart algebras, descent, and quantization
• Stacky Lie groups
• Atiyah classes of Lie bialgebras
• Integrability of Poisson brackets
• Equivariant resolutions of singularities as differential substacks

Cited by in corpus (2)

• Atiyah classes of strongly homotopy Lie pairs
• Classification of Group Actions Extended to Symplectic Deformation Quantizations