From Atiyah Classes to Homotopy Leibniz Algebras
Zhuo Chen, Mathieu Sti\'enon, Ping Xu

TL;DR
This paper generalizes Kapranov's theorem, showing that Atiyah classes induce Lie algebra and homotopy Leibniz algebra structures in derived categories for vector bundles over Lie pairs, extending classical results.
Contribution
It extends Kapranov's theorem to Lie pairs, defining Atiyah classes that induce Lie and homotopy Leibniz algebra structures in broader contexts.
Findings
Atiyah classes induce Lie algebra structures in derived categories.
Construction of homotopy Leibniz algebra from Atiyah classes.
Generalization of classical theorems to Lie pairs and modules.
Abstract
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold makes into a Lie algebra object in , the bounded below derived category of coherent sheaves on . Furthermore Kapranov proved that, for a K\"ahler manifold , the Dolbeault resolution of is an 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 , i.e. a Lie algebroid together with a Lie subalgebroid , we define the Atiyah class of an -module (relative to ) as the obstruction to the existence of an -compatible -connection on . We prove that the Atiyah classes and respectively make and into a Lie algebra and a Lie algebra module in…
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