From Atiyah Classes to Homotopy Leibniz Algebras

Zhuo Chen Tsinghua University Mathieu Stienon Penn State Univerisity Ping Xu Penn State University

Mathematical Physics mathscidoc:1702.22013

Communications in Mathematical Physics , 341, (1), 309-349, 2016
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 \Omega(T{1,0} X ) 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 Liealgebra 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 afore said Lie structures in D+(A).
Atiyah Class, Homotopy Lie Algebra, Homotopy Leibniz Algebra, Lie Pair,
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  title={From Atiyah Classes to Homotopy Leibniz Algebras},
  author={Zhuo Chen, Mathieu Stienon, and Ping Xu},
  booktitle={Communications in Mathematical Physics  },
Zhuo Chen, Mathieu Stienon, and Ping Xu. From Atiyah Classes to Homotopy Leibniz Algebras. 2016. Vol. 341. In Communications in Mathematical Physics . pp.309-349.
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