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 X(M) on a dg-manifold M with homological vector field Q admits a structure of L∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.