Abstract
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 \({D^+(\mathcal{A})}\) .
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Communicated by N. Reshetikhin
Research partially supported by NSFC Grant 11471179, the Beijing high education young elite teacher project, NSA Grant H98230-12-1-0234, and NSF Grants DMS0605725, DMS0801129, DMS1101827.
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Chen, Z., Stiénon, M. & Xu, P. From Atiyah Classes to Homotopy Leibniz Algebras. Commun. Math. Phys. 341, 309–349 (2016). https://doi.org/10.1007/s00220-015-2494-6
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DOI: https://doi.org/10.1007/s00220-015-2494-6