Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

Abstract

We investigate vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair (L,A), i.e. a pair of a Lie algebroid L and a Lie subalgebroid A of L. The construction of Fedosov dg manifolds involves a choice of a splitting and a connection. We prove that, given any two choices of a splitting and a connection, there exists a unique vertical isomorphism, determined by an iteration formula, between the two associated Fedosov dg manifolds. As an application, we provide an explicit formula for the map pbw2−1∘pbw1 associated with two Poincaré–Birkhoff–Witt isomorphisms that arise from two choices of a splitting and a connection.