Global Lie-Tresse theorem

Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfeld

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra associated to the Poisson Lie group is proved to hold in this more general setting.

The Generalized Stokes theorem for R-linear forms on Lie algebroids

The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.

伪黎曼Hom-李代数中的Killing向量与Bochner型定理

Killing Vectors and Bochner-Type Theorems in Pseudo-Riemannian Hom-Lie Algebras

研究伪黎曼Hom-李代数，建立黎曼和洛伦兹情形下经典Bochner定理的Hom-版本，并证明Killing向量空间构成全测地Hom-李子代数。

李群胚 Riemann-Roch-Hirzebruch 定理及应用

Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

本文引入李代数胚指标理论并研究其 Dolbeault 算子，结合正则叶状流形上的 Connes 指标理论建立广义 Riemann-Roch 定理，并提出 Kodaira 消灭定理以通过拓扑信息验证正线丛判据。

Cartan's theorem for some topological generalized groups

In this paper we show that topological subgroupoids of Lie groupoids, under special circumstances are Lie subgroupoids. Giving an example, we indicate that having the same topological dimension is a necessary condition for topological subgroupoids to be Lie subgroupoids. Also, we provide some conditions for double subgroupoids to become double Lie subgroupoids. Moreover, we illustrate that having the same conditions as the Cartan's theorem for Lie groups, helps us prove the same theorem for generalized Lie groups.

Kähler李代数胚上的Hard Lefschetz定理

The Hard Lefschetz Theorem on Kähler Lie Algebroids

本文证明了在椭圆性条件下，一类Kähler李代数胚满足Hard Lefschetz定理，并给出了满足与不满足该条件的例子。

Kähler Lie algebroids 的 Kodaira 消失定理

Kodaira vanishing theorems for Kahler Lie algebroids

本文将复几何中的 Kodaira 消失定理推广到 Kähler Lie algebroids，证明当 p+q 足够大时正线丛值 (p,q)-形式上的 Laplace 算子核消失，并利用局部坐标计算推广 Kähler 恒等式。

Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense that the bracket relations among them can be written with relatively smaller number of parameters. Our procedure is based on the moduli space of left-invariant Riemannian metrics. Some explicit examples of such frames and applications will also be given.

Half-space theorems and the embedded Calabi-Yau problem in Lie groups

We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfaces. This half-space theorem applies to certain properly immersed constant mean curvature surfaces of X contained in the complements of normal R^2 subgroups F of X. In the case X is a unimodular Lie group, our results imply that every minimal surface in X-F that is properly immers…

m > 1因变量的牛顿自由质点系的显式微分特征

Explicit differential characterization of the Newtonian free particle system in m > 1 dependent variables

1883年，作为早期的结果，Sophus Lie建立了一个显式的充要条件，使得解析二阶常微分方程y_xx = F（x，y，y_x）通过点变换（x，y）-->（X（x，y），Y（x，y））等价于牛顿自由粒子方程Y_XX = 0。这个结果是二阶解析常微分方程的群论分类的初步结果，后来在1896年由S.撒谎本文遵循S. Lie，我们首先暴露和恢复长度，我们推广到几个二阶常微分方程的情况下，这一明确的特征。设K=R或C，或者更一般地，设任何特征为零的域具有赋值，这样K-解析函数就有意义了。令x在K中，令m > 1，令y：=（y^1，...，y^m）在K^m中，并且令y_xx^j = F^j（x，y，y_x^l），j = 1，.，m是m个解析二阶常微分方程的集合，一般是非线性的。我们给出了一个显式的充要条件，以使该系统在点变换（x，y^1，.，y^m）-->（X（x，y），Y ^1（x，y），.，Y^m（x，y）），到牛顿自由粒子系统Y_XX^1 =. = Y_XX^m = 0。引人注目的是，我们得到的（复杂）微分系统在m > 1的情况下是一阶的，而在S中是二阶的。李的原始情况m = 1。

Lie II theorem for Lie algebroids via higher groupoids

Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale Lie 2-groupoids with 2-connected source fibres. Finally, we discuss how to lift Lie algebroid morphisms to Lie 2-groupoid morphisms (Lie II Theorem).

The Frobenius theorem for Banach distributions on infinite-dimensional manifolds and applications in infinite-dimensional Lie theory

We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex space that admits an exponential map, the Lie subalgebra is integrable if it is complemented as a topological vector space and a Banach space with the induced topology.

李代数胚的辛约化与 Darboux-Moser-Weinstein 定理

Symplectic reduction and a Darboux-Moser-Weinstein theorem for Lie algebroids

本文将 Marsden-Weinstein 约化定理与 Darboux-Moser-Weinstein 定理推广至辛李代数胚，并提出了相应的余迷向嵌入定理。

关于半单李群的Livsic定理、超刚性和Anosov作用

On Livsic's Theorem, Superrigidity, and Anosov Actions of Semisimple Lie Groups

证明了Livsic定理关于某些类型动力系统的上同调消失的推广。因此，我们加强了Zimmer关于半单李群的Anosov作用的代数壳的一个结果。结合拓扑超刚性，我们找到了多重自由Anosov作用的Holder几何结构。

Lie theorem via rank 2 distributions (integration of PDE of class ω=1)

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, as in the method of Darboux, and discuss nonlinear Laplace transformations and symmetric PDE models.

A Pluzhnikov's Theorem, Brownian motions and Martingales in Lie Group with skew-symmetric connections

Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem.

A Dixmier-Malliavin theorem for Lie groupoids

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from…

A PBW theorem for inclusions of (sheaves of) Lie algebroids

Inspired by the recent work of Chen-Stiénon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincaré-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.

The Nash-Moser Theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\mathfrak{g},\mathfrak{g})$, and certain subspaces of it, that provide fine information about the deformations of $\mathfrak{g}$ in the variety of $k$-step nilpotent Lie algebras. Then we focus on degenerations and rigidity in the variety of $k$-step nilpotent Lie algebras of dimensio…

Generalized Lie-Bäcklund theorem for Lie class ω=1 overdetermined systems

In this paper we prove a version of Lie-Bäcklund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the plane treated by E.Cartan in 1910. Many examples are provided.

Categorified central extensions, étale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras

Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they a…

Lie第三定理的Lie ∞-代数推广

Lie's Third Theorem for Lie $\infty$-Algebras

本文通过局部最小模型理论，首次证明了有限型Lie ∞-代数的Lie第三定理，并给出了弦Lie 2群的新模型。

On the linearization theorem for proper Lie groupoids

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).

A Serre-Swan theorem for gerbe modules on étale Lie groupoids

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.

The localized longitudinal index theorem for Lie groupoids and the van Est map

We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition as well as a computation of the "global …

The transverse index theorem for proper cocompact actions of Lie groupoids

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.

A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.

Interior and Exterior Differential Systems for Lie Algebroids

A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending the classical notion of exterior differential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.

传递李代数胚上外微分系统的Cartan-Kähler定理

The Cartan-Kähler theorem for exterior differential systems on transitive Lie algebroids

将外微分系统推广到李代数胚，在锚映射满射条件下给出两个Cartan-Kähler定理版本，并应用于变分学的不变逆问题。

李群齐性空间的指标定理

Index theorem for homogenous spaces of Lie groups

本文研究几乎连通李群齐性空间上的指标理论，分别从拓扑与分析两方面出发，利用Riemann-Roch公式和热核方法得到拓扑公式与局部指标公式。

量化版本的潘苏渐近定理和米切尔坦格定理

Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem

本文量化研究了测地黎曼群向其度量极限收敛的速度，并改进了现有文献中的相关界。

Lie Groupoids in Classical Field Theory II: Gauge Theories, Minimal Coupling and Utiyama's Theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. In this second part, we shall adapt the formalism developed in the first paper to the context of gauge theories and deal with minimal coupling and Utiyama's theorem.

Lie Groupoids in Classical Field Theory I: Noether's Theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether's theorem about the connection between symmetries and conservation laws in this approach.