TY - JOUR

T1 - From Atiyah classes to homotopy Leibniz algebras

AU - Chen, Zhuo

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information:
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.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.

PY - 2016/1

Y1 - 2016/1

N2 - 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).

AB - 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).

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U2 - 10.1007/s00220-015-2494-6

DO - 10.1007/s00220-015-2494-6

M3 - Article

AN - SCOPUS:84952989197

VL - 341

SP - 309

EP - 349

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -