### 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 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).

Original language | English (US) |
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Pages (from-to) | 309-349 |

Number of pages | 41 |

Journal | Communications In Mathematical Physics |

Volume | 341 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*341*(1), 309-349. https://doi.org/10.1007/s00220-015-2494-6