From Atiyah Classes to Homotopy Leibniz Algebras

Details From Atiyah Classes to Homotopy Leibniz Algebras From Atiyah Classes to Homotopy Leibniz Algebras

2012-04-04

arxiv.org/abs/1204.1075v1

Mathematics

Differential Geometry

34 pages

Scientific paper

Given a Lie pair $(L,A)$, i.e.\ a Lie algebroid $L$ together with a Lie subalgebroid $A$ (whose sheaf of smooth sections is noted $\mathscr{A}$), we define the Atiyah class $\alpha_E$ of an $A$-module $E$ as the obstruction to the existence of an \emph{$A$-compatible} $L$-connection on $E$. We prove that the Atiyah classes $\alpha_{L/A}$ and $\alpha_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the derived category $D^+(\mathcal{A})$, where $\mathcal{A}$ is the category of coherent sheaves of $\mathscr{A}$-modules. This generalizes a result of Kapranov concerning the classical Atiyah class of a holomorphic vector bundle. 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 $D^+(\mathcal{A})$.

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