Fedosov dg manifolds associated with Lie pairs

TL;DR

This paper constructs Fedosov dg manifolds from Lie algebroid pairs, linking the Fedosov iteration method with the Poincaré--Birkhoff--Witt map and establishing a homotopy equivalence between associated dg algebras.

Contribution

It introduces a new class of Fedosov dg manifolds for Lie algebroid pairs and connects their structure to existing algebraic maps, providing a homotopy equivalence result.

Findings

Construction of Fedosov dg manifolds from Lie pairs.

Connection of the cohomological vector field to the PBW map.

Establishment of a homotopy equivalence between dg algebras.

Abstract

Given any pair $(L,A)$ of Lie algebroids, we construct a differential graded manifold $(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the cohomological vector field $Q$ constructed on $L[1]\oplus L/A$ by the Fedosov iteration method arises as a byproduct of the Poincar\'e--Birkhoff--Witt map established in arXiv:1408.2903. Finally, using the homological perturbation lemma, we establish a quasi-isomorphism of Dolgushev--Fedosov type: the differential graded algebras of functions on the dg manifolds $(A[1],d_A)$ and $(L[1]\oplus L/A,Q)$ are homotopy equivalent.