2012/04/30 by Zhuo Chen, Mathieu Stiénon, Ping Xu
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bounded function #Graded Lie algebra #Homotopy #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Mathematical analysis #Mathematics #Pure mathematics #Tangent bundle #Tangent space #Vector bundle #hep-th #math.AG #math.DG #math.QA
paper · pdf · doi:10.1007/s00220-015-2494-6
published as Comm. Math. Phys. 341 (2016), no. 1, 309-349 · 36 pages
arxiv created 2015/09/11 · openalex publication_date 2015/12/01 · arxiv updated 2017/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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\bullet-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 (relative to L) 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).