Fedosov dg manifolds associated with Lie pairs

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 Poincaré--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.

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References in corpus (4)

• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Atiyah classes and dg-Lie algebroids for matched pairs
• Formality for g-manifolds
• Formal exponential map for graded manifolds

Cited by in corpus (3)

• Hochschild cohomology of dg manifolds associated to integrable distributions
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Hopf algebras arising from dg manifolds