The Atiyah class of a dg-vector bundle

Abstract

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary bracket, and the Atiyah cocycle corresponding to a torsion-free affine connection as binary bracket.

7 pages. To appear in Compte Rendus Math

References in corpus (7)

• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
• Rozansky-Witten invariants via Atiyah classes
• Double Lie algebroids and the double of a Lie bialgebroid
• Q-algebroids and their cohomology
• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• Characteristic classes associated to Q-bundles
• On the Duflo formula for $L_{\infty }$-algebras and Q-manifolds

Cited by in corpus (22)

• From Atiyah Classes to Homotopy Leibniz Algebras
• Introduction to graded geometry
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Atiyah classes and dg-Lie algebroids for matched pairs
• Formality for g-manifolds
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Formal exponential map for graded manifolds
• Atiyah and Todd classes arising from integrable distributions
• Derived Differentiable Manifolds
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Hopf algebras arising from dg manifolds
• Basic curvature and the Atiyah cocycle in gauge theory
• Multiplicative Connections and Their Lie Theory
• Atiyah classes of strongly homotopy Lie pairs
• Polyvector fields and polydifferential operators associated with Lie pairs
• Keller admissible triples and Duflo theorem
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• $A_{\infty }$-Algebras from Lie Pairs
• The Atiyah class of generalized holomorphic vector bundles
• Atiyah and Todd classes of regular Lie algebroids
• Dg Loday-Pirashvili modules over Lie algebras