arXiv math/07025842007-02math.AGmath.DG

Free Lie algebroids and the space of paths

Mikhail Kapranov

备注：: 42 pages, revised version, to appear in Selecta Math

英文摘要：

We construct algebraic and algebro-geometric models for the spaces of unparametrized paths. This is done by considering a path as a holonomy functional on indeterminate connections. For a manifold X, we construct a Lie algebroid P which serves as the tangent space to X (punctual paths) inside the space of all unparametrized paths. It serves as a natural receptacle of all "covariant derivatives of the curvature" for all bundles with connections on X. If X is an algebraic variety, we integrate P to a formal groupoid G which can be seen as the formal neighborhood of X inside the space of paths. We establish a relation of G with the stable map spaces of Kontsevich.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Integrating Lie algebroids via stacks

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into a Weinstein groupoid.

2006-11-27math.AGmath.DG

抛物向量束与李代数丛连接

Parabolic vector bundles and Lie algebroid connections

本文研究了抛物向量束与李代数丛连接的关系，构造了抛物李代数丛的Atiyah序列，并对稳定李代数的抛物向量束进行了特征化。

2025-12-22math.AGmath.DG

Invariants of Lie algebroids

Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.

2002-06-10math.AGmath.DG

关于抛物向量束 admits 抛物李代数束连接的准则

A criterion for parabolic vector bundles to admit a parabolic Lie algebroid connection

本文研究了抛物向量束在给定抛物结构下是否能配备抛物李代数束连接的条件，通过分析给出了必要且充分的判断标准。

2026-04-29math.AGmath.DG

全纯李代数丛联络的一个判据

A criterion for holomorphic Lie algebroid connections

该研究针对紧连通黎曼曲面上的全纯李代数丛，给出全纯向量丛容许其全纯联络的充要条件，区分非分裂与分裂情形分别得到对应结论。

2025-06-12math.AGmath.DG

李代数、量子泊松代数以及李代数连接

Lie algebroids, quantum Poisson algebroids, and Lie algebroid connections

本文研究了覆盖环面的李代数与量子泊松sheaf之间的双射对应关系，并探讨了非平坦李代数连接的构造方法。

2025-12-10math.AG

更多相关论文收起更多论文

Lie algebroids as $L_\infty$ spaces

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_\infty$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_\infty$ space. Indeed, this functor sends the …

2018-01-18math.AGmath.DGmath.QA

Families and unfoldings of singular holomorphic Lie Algebroids

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.

2021-12-29math.AGmath.DGmath.SG

Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.

2017-11-14math.AGmath.RA

Koszul duality for Lie algebroids

This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.

2017-12-09math.AGmath.AT

Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O $ and homotopy equivalence classes of negatively graded Lie $\infty $-algebroids over their resolutions (=acyclic Lie $\infty$-algebroids). This extends to a purely algebraic setting the construction of the universal $Q$-manifold of a locally real analytic singular foliation of Lavau-C.L.-Strobl. In particular, it makes sense for the universal Lie $\infty$-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algeb…

2021-11-25math.AGmath.AT

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

2014-11-20math.AGmath.SG