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 ${\mathrm{pbw}}_2^{-1}\circ {\mathrm{pbw}}_1$ associated with two Poincaré--Birkhoff--Witt isomorphisms that arise from two choices of a splitting and a connection.

22 pages. Minor revision; some terminologies changed; some proofs improved. To appear in Journal of Geometry and Physics