From Atiyah Classes to Homotopy Leibniz Algebras

Abstract

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ 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 ${\Omega }^{\bullet -1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_{\infty }$ 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 ${\alpha }_E$ 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 ${\alpha }_{L/A}$ and ${\alpha }_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^{+}(\mathcal{A})$, where $\mathcal{A}$ is the abelian category of left $\mathcal{U}(A)$-modules and $\mathcal{U}(A)$ 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 $D^{+}(\mathcal{A})$.

36 pages

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Cited by in corpus (31)

• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• On the Strong Homotopy Associative Algebra of a Foliation
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Shifted derived Poisson manifolds associated with Lie pairs
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Atiyah classes and dg-Lie algebroids for matched pairs
• Formality for g-manifolds
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Atiyah and Todd classes arising from integrable distributions
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Invariant connections and PBW theorem for Lie groupoid pairs
• A Hopf algebra associated to a Lie pair
• Fedosov dg manifolds associated with Lie pairs
• Basic curvature and the Atiyah cocycle in gauge theory
• Hopf algebras arising from dg manifolds
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Polyvector fields and polydifferential operators associated with Lie pairs
• Atiyah classes of strongly homotopy Lie pairs
• The Dolbeault dga of the formal neighborhood of the diagonal
• Infinitesimal ideal systems and the Atiyah class
• $A_{\infty }$-Algebras from Lie Pairs
• The Atiyah class of generalized holomorphic vector bundles
• On higher order Leibniz identities in TCFT
• Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid
• Atiyah and Todd classes of regular Lie algebroids
• Dg Loday-Pirashvili modules over Lie algebras
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair
• Characteristic classes of Higgs bundles and Reznikov's theorem
• On Lie Bialgebroid Crossed Modules
• The $\hat{A}$ genus as a projective volume form on the derived loop space
• Obstructions to representations up to homotopy and ideals