A Lie pair (L,A) consists of a Lie algebra (or more generally, a Lie algebroid) L
together with a Lie subalgebra (or Lie subalgebroid) A.
A wide range of geometric situations can be described in terms of Lie pairs
including complex manifolds, foliations, and manifolds equipped with Liie group actions.
To each Lie pair (L,A) are associated two L-infinity algebras,which play roles similar to the spaces of polyvector fields and polydifferential operators.
We establish the formality theorem for Lie pairs. As an application, we obtain Kontsevich-Duflo type theorem for Lie pairs.
Besides using Kontsevich formality theorem, our approach is based on the construction of a dg manifold (L + L/A, Q) together with a dg foliation,
called the Fedosov dg Lie algebroid.
This is a joint work with Hsuan-Yi Liao and Mathieu Stienon.