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 Kähler manifold X, the Dolbeault resolution Ω^•−1(T_X^1,0) of T_X [−1] is an L_∞ 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 α_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 α_L/A and α_E respectively make L/A[−1] and E[−1] into a Lie algebra and a Lie algebra module in the bounded below derived category D⁺(A), where A is the abelian category of left U(A)-modules and U(A) 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 D⁺(A).