Hopf algebras arising from dg manifolds

TL;DR

This paper constructs a Hopf algebra structure from the universal enveloping algebra of a Lie algebra object associated with a dg manifold, linking dg geometry with algebraic structures.

Contribution

It describes the universal enveloping algebra of a Lie algebra object in a homotopy category and proves it forms a Hopf algebra, extending dg geometric structures.

Findings

Universal enveloping algebra forms a Hopf algebra in the homotopy category.

Connects dg manifolds with algebraic Hopf structures.

Recovers known results on Hopf algebras from Lie pairs.

Abstract

Let $(\mathcal{M}, Q)$ be a dg manifold. The space of vector fields with shifted degrees $(\mathcal{X}(\mathcal{M})[-1], L_Q)$ is a Lie algebra object in the homology category $\mathrm{H}((C^{\infty}_{\mathcal{M}},Q)\mathrm{-}\mathbf{mod})$ of dg modules over $(\mathcal{M},Q)$, the Atiyah class $\alpha_{\mathcal{M}}$ being its Lie bracket. The triple $(\mathcal{X}(\mathcal{M})[-1], L_Q; \alpha_{\mathcal{M}})$ is also a Lie algebra object in the Gabriel-Zisman homotopy category $\Pi((C^{\infty}_{\mathcal{M}},Q)\mathrm{-}\mathbf{mod})$. In this paper, we describe the universal enveloping algebra of $(\mathcal{X}(\mathcal{M})[-1], L_Q; \alpha_{\mathcal{M}})$ and prove that it is a Hopf algebra object in $\Pi((C^{\infty}_{\mathcal{M}},Q)\mathrm{-}\mathbf{mod})$. As an application, we study Fedosov dg Lie algebroids and recover a result of Sti\'enon, Xu, and the second author on the Hopf algebra arising from a Lie pair.