From Atiyah classes to homotopy Leibniz algebras

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A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes TX [−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(TX1,0) of TX [−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 αE 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 αL/A and αE 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).

Original languageEnglish (US)
Pages (from-to)309-349
Number of pages41
JournalCommunications In Mathematical Physics
Issue number1
StatePublished - Jan 2016

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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