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 \({\Omega^{\bullet-1}(T_X^{1, 0})}\) 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 \({D^+(\mathcal{A})}\) , where \({\mathcal{A}}\) is the abelian category of left \({\mathcal{U}(A)}\) -modules and \({\mathcal{U}(A)}\) 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 \({D^+(\mathcal{A})}\) .