Hopf algebras arising from dg manifolds

Let (M, Q) be a dg manifold. The space of vector fields with shifted degrees (X(M)[-1], L-Q) is a Lie algebra object in the homology category H((C-M(infinity), Q)-mod) of dg modules over (M, Q), the Atiyah class alpha(M) being its Lie bracket. The triple (X(M)[-1], L-Q; alpha(M)) is also a Lie algebra object in the Gabriel-Zisman homotopy category Pi((C-M(infinity), Q)-mod). In this paper, we describe the universal enveloping algebra of (X(M)[-1], L-Q; alpha(M)) and prove that it is a Hopf algebra object in Pi((C-M(infinity), Q)-mod). As an application, we study Fedosov dg Lie algebroids and recover a result of Stienon, Xu, and the second author on the Hopf algebra arising from a Lie pair. (C) 2021 Elsevier Inc. All rights reserved.