From Atiyah Classes to Homotopy Leibniz Algebras

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• 作者：Zhuo Chen ; Mathieu Stiénon ; Ping Xu
• 刊名：Communications in Mathematical Physics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：341
• 期：1
• 页码：309-349
• 全文大小：776 KB
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• 作者单位：Zhuo Chen (1)
  Mathieu Stiénon (2)
  Ping Xu (2)
  
  1. Department of Mathematics, Tsinghua University, Beijing, China
  2. Department of Mathematics, Pennsylvania State University, University Park, PA, USA
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

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 \({D^+(\mathcal{A})}\) . 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.Communicated by N. Reshetikhin