Rozansky–Witten-Type Invariants from Symplectic Lie Pairs

Voglaire, Yannick; Xu, Ping

doi:10.1007/s00220-014-2221-8

Cited by 6 publications

(5 citation statements)

References 27 publications

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“…As shown by the pioneering work of Kapranov [24,41,42], the Atiyah class of a holomorphic vector bundle gives rise to L ∞ [1] algebras. These L ∞ [1] algebras play an important role in derived geometry [10,35,41] and the construction of Rozansky-Witten invariants [24,26,42,51].…”

Section: Our First Main Theorem Confirms This Assertionmentioning

confidence: 99%

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Dg manifolds, formal exponential maps and homotopy Lie algebras

Seol,

Stiénon,

2021

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This paper is devoted to the study of the relation between 'formal exponential maps,' the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C ∞ context. Given a dg manifold, we prove that a 'formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L∞[1] algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold (T 0,1 X [1], ∂) arising from a complex manifold X, we prove that this L∞[1] algebra structure is quasi-isomorphic to the standard L∞[1] algebra structure on the Dolbeault complex Ω 0,• (T 1,0 X ).

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Section: Our First Main Theorem Confirms This Assertionmentioning

confidence: 99%

“…The Atiyah class of a holomorphic vector bundle is closely related to L ∞ [1] algebras as shown by the pioneer work of Kapranov [24,41,42]. These L ∞ [1] algebras play an important role in derived geometry [10,35,41] and construction of Rozansky-Witten invariants [24,26,42,51].…”

Section: Atiyah Class and Homotopy Lie Algebrasmentioning

confidence: 99%

Dg manifolds, formal exponential maps and homotopy Lie algebras

Seol,

Stiénon,

2021

Preprint

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“…The universal enveloping algebra of the Lie algebra object L/A[−1] in the derived category D + (A) (see Theorem B) is described in that same paper [13]. We note that the Atiyah class of Lie algebroid pairs plays a central role in the construction of new Rozansky-Witten type invariants of 3-manifolds from symplectic Lie pairs [54]. In another direction, Atiyah classes were defined for differential graded vector bundles and these yield homotopy algebraic structures as well [39].…”

Section: We Call the Extension Classmentioning

confidence: 99%

“…Molino's class has applications in geometry, for instance, in the study of differential operators on foliated manifolds [50] and in deformation quantization [6]. This paper is the first in a sequence of works [13,31,32,54,39] which aim at developing in a general setting a theory of Atiyah classes and their applications. Our goal is to explore emerging connections between derived geometry and classical areas of mathematics such as complex geometry, foliation theory, Poisson geometry and Lie theory.…”

Section: Introductionmentioning

confidence: 99%

From Atiyah Classes to Homotopy Leibniz Algebras

Chen

Stiénon

2015

Commun. Math. Phys.

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105

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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

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“…In the last decade much research on Lie pairs has been done following different strategies and the underlying mathematical structures: Atiyah classes arising from Lie pairs have been studied, using a variety of methods, see e.g. [2,8,9]; It is shown that geometric objects including Kapranov dg and Fedosov dg manifolds [19,30], algebraic objects such as Hopf algebras [7,11], Leibniz ∞ and L ∞ algebras can be derived from Lie pairs [1,6,20]; Also, in the context of Lie pairs, considerable attentions had been paid to Poincaré-Birkhoff-Witt isomorphisms [4,5], Kontsevich-Duflo isomorphisms [10,21], and Rozansky-Witten-type invariants [37], etc.…”

Section: Introductionmentioning

confidence: 99%

Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

Ni¹,

Cheng²,

Chen³

et al. 2022

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A Lie pair is an inclusion A to L of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical L 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(L/A) whose unary bracket is the Chevalley-Eilenberg differential arising from every Lie pair (L, A). In this note, we prove that to such a Lie pair there is an associated Lie algebra action by Diff(L) on the L 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(L/A). Here Diff(L) is the space of 1-differentials on the Lie algebroid L, or infinitesimal automorphisms of L. The said action gives rise to a larger scope of gauge equivalences of Maurer-Cartan elements in Γ(∧ • A ∨ ) ⊗ Γ(L/A), and for this reason we elect to call the Diff(L)-action internal symmetry of Γ(∧ • A ∨ ) ⊗ Γ(L/A).

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Rozansky–Witten-Type Invariants from Symplectic Lie Pairs

Cited by 6 publications

References 27 publications

Dg manifolds, formal exponential maps and homotopy Lie algebras

Dg manifolds, formal exponential maps and homotopy Lie algebras

From Atiyah Classes to Homotopy Leibniz Algebras

Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

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