Hopf algebras arising from dg manifolds

Let (M,Q) be a dg manifold. The space of vector fields with shifted degrees (X(M)[−1],LQ) is a Lie algebra object in the homology category H((CM∞,Q)−mod) of dg modules over (M,Q), the Atiyah class αM being its Lie bracket. The triple (X(M)[−1],LQ; αM) is also a Lie algebra object in the Gabriel-Zisman homotopy category Π((CM∞,Q)−mod). In this paper, we describe the universal enveloping algebra of (X(M)[−1],LQ;αM) and prove that it is a Hopf algebra object in Π((CM∞,Q)−mod). As an application, we study Fedosov dg Lie algebroids and recover a result of Stiénon, Xu, and the second author on the Hopf algebra arising from a Lie pair.