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 K\"ahler 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 $D^+(\mathcal{A})$.Comment: 36 page

A Hopf algebra associated with a Lie pair

“…Choose an L-connection ∇ on L/A extending the A-action. Its curvature is the vector bundle map The following was proved in [1].…”

“…a Lie algebroid L with a Lie subalgebroid A, the Atiyah class α E of an A-module E relative to the pair (L, A) is defined as the obstruction to the existence of an A-compatible L-connection on the vector bundle E. An L-connection ∇ on an A-module E is said to be A-compatible if it extends the given flat A-connection on E and satisfies ∇ a ∇ l − ∇ l ∇ a = ∇ [a,l] for all a ∈ Γ(A) and l ∈ Γ(L). This fairly recently defined class (see [1]) has as double origin, which it generalizes, the Atiyah class of holomorphic vector bundles and the Molino class of foliations.…”

“…The quotient L/A of any Lie pair (L, A) is an A-module [1]. Its Atiyah class α L/A can be described as follows.…”

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

“…We suspect, based on this, that there may exist a deep link between solutions of the generalized Kashiwara-Vergne problem [1] and the above isomorphisms between harmonic and Hochschild structures on an algebraic variety. It is likely that this link can be found using the ideas developed in [10], [11], [14], [37], and [40].…”

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

“…Kapranov's construction of L ∞ algebras is generalized in Chen, Stiénon and Xu's work [8] where the setting is a Lie algebroid pair (Lie pair, for short) (L, A). It is shown that the graded vector space Γ(∧ • A ∨ ⊗ L/A) admits a Leibniz ∞ [1] algebra structure ([8, Theorem 3.13]) via the Atiyah class of the Lie pair (L, A).…”