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

References in corpus (22)

• Deformation quantization of Poisson manifolds, I
• The Geometry of the Master Equation and Topological Quantum Field Theory
• Geometry of Batalin-Vilkovisky quantization
• Quantum groupoids
• Hochschild cohomology and Atiyah classes
• From Atiyah Classes to Homotopy Leibniz Algebras
• On the Rozansky-Witten weight systems
• The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
• Rozansky-Witten invariants via Atiyah classes
• The Atiyah class of a dg-vector bundle
• Rozansky-Witten invariants via formal geometry
• On the Strong Homotopy Associative Algebra of a Foliation
• One-dimensional Chern-Simons theory and the $\hat{A}$ genus
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Shifted derived Poisson manifolds associated with Lie pairs
• 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
• Kapranov's construction of sh Leibniz algebras
• Fedosov dg manifolds associated with Lie pairs
• A Hopf algebra associated to a Lie pair

Cited by in corpus (3)

• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair