From Atiyah Classes to Homotopy Leibniz Algebras

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T (X) [-1] into a Lie algebra object in D (+) (X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kahler manifold X, the Dolbeault resolution of T (X) [-1] is an L (a) algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair (L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class alpha (E) of an A-module E as the obstruction to the existence of an A-compatible L-connection on E. We prove that the Atiyah classes alpha (L/A) and alpha (E) respectively make L/A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the bounded below derived category , where is the abelian category of left -modules and is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/A and E, and inducing the aforesaid Lie structures in .