Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

TL;DR

This paper proves the existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds linked to a Lie pair, showing how different choices of splittings and connections relate explicitly.

Contribution

It establishes a unique vertical isomorphism between Fedosov dg manifolds for any two choices of splittings and connections, with an explicit formula for the associated PBW maps.

Findings

Existence of a unique vertical isomorphism between Fedosov dg manifolds.

Explicit formula for the map between PBW isomorphisms.

Application to Lie pairs and their associated geometric structures.

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 Poincar\'{e}--Birkhoff--Witt isomorphisms that arise from two choices of a splitting and a connection.