Hopf algebras arising from dg manifolds

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 $\mathrm{H}((C_{\mathcal{M}}^{\infty },Q)-\mathbf{mod})$ 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 $\Pi ((C_{\mathcal{M}}^{\infty },Q)-\mathbf{mod})$. 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 $\Pi ((C_{\mathcal{M}}^{\infty },Q)-\mathbf{mod})$. 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.

37 pages, published in Journal of Algebra