TY - JOUR
T1 - From Atiyah classes to homotopy Leibniz algebras
AU - Chen, Zhuo
AU - Stiénon, Mathieu
AU - Xu, Ping
N1 - 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
SN - 0010-3616
VL - 341
SP - 309
EP - 349
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 1
ER -