Linus Kramer
Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.
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The core diagram of a double Lie algebroid consists in the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram. The comma double Lie algebroid associated to a morphism of Lie…
Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given ac…
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affi…
Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.
Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we no…
On post-Lie algebras structures coming from simply transitive NIL-affine actions
本文证明了单连通可解 Lie 群在幂零 Lie 群上的单传递 NIL-仿射作用可诱导 post-Lie 代数结构，并在 2 步幂零情形下确立了单传递作用与完备 post-Lie 代数结构的精确对应。
On the integration of transitive Lie algebroids
针对可递李代数胚积分到李群胚的问题，给出Crainic-Fernandes障碍的几何解释及障碍消失时的显式构造方法，并提出向正则李代数胚扩展的思路。
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. This paper proves the existence of suitably equivariant transition functions for such groupoids, generalising consequently the classification of principal bundles by means of their transition functions, to extensions of princip…
We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others. This paper gives just one, subjective overview of the subject, without trying to be exhaustive.
In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.
We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ vanishes unless $A=TM$, and prove a general Künneth formula. As applications we give a short proof of a vanishing result for the Euler characteristic of a principal bundle calculated using invariant differential forms, and show that the cohomology of certain Lie algebroids are exterior algebras. The latter result can be seen as a generalization o…
The preprint is prepared as description of results that were obtained during joint scientific project No: 71NC /2015/VNCCCT on the VIASM (Vietnam Institute for Advanced Study in Mathematics) from 08.12.2015 to 06.02.2016. The problem was formulated how to calculate so called the Mackenzie obstruction for existing of transitive Lie algebroid for given coupling between a finite dimensional Lie algebra and the tangent bundle of a smooth manifold. It is proved that the Mackenzie obstruction for transitive Lie algebroids is trivial for the finite dimensional Lie algebra which is the direct sum of t…
Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This family has been introduced recently and possesses interesting geometric properties. If $p=q=0$ we recover the Sasaki metric and when $p=q=1$ we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean prod…
Transitive Lie algebroids and \textbf{Q}-manifolds
引入局部平凡\textbf{Q}-群胚和主\textbf{Q}-丛概念，证明每个传递李代数胚可由某个主\Q-丛的Atiyah序列得到，从而被局部平凡\textbf{Q}-群胚积分。
Algebraic Integrability and Minimality of Lie Equations for Transitive, Finite-Dimensional, Non-Commutative Lie Pseudogroups
该研究通过微分伽罗华理论，探讨了D-群oids的代数可积性与极小性，揭示了其与微分伽罗华群的关系。
Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids
本文探讨如何为1阶余秩子黎曼流形（包括接触和准接触子黎曼流形）构造合适的非传递卡坦连接，通过将子黎曼符号的变化转化为合适的
For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.
This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. We encode geometric structures induced by transitive Lie pseudogroups into principal $G$-bundles equipped with a transversally parallelisable foliation generated by a subalgebra of $\mathfrak{g}$, called Cartan bundles. Our approach is complementary to arXiv:1911.13147 and is based on Morita equivalence of Lie groupoids. After identifying the main examples and properties, we develop a notion of flatness with respect…
In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to o…
We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable and transitive Lie algebras of vector fields whose derivative ideal is nilpotent can be adapted to this scheme.
Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms
本文提出一种仅基于李理论标准结论的李代数方法，为紧复齐性流形相关经典结论给出新证明，还提供了将典范阿贝尔李代数与紧李代数上可积复结构关联的方式，拓展了前人研究。
On the equivalence of the integrability obstructions for transitive Lie algebroids
本文探讨了传递李叠映射积分性障碍的不同视角，揭示了同调方法与同调构造的相互作用，证明了Mackenzie的同调障碍与Crainic和Fernandes的单胞映射以及Meinrenken的粘合构造等价。
In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on the group $\Aut(\rg)$ of all automorphisms of the Lie algebra $\rg$, say $\Aut(\rg)^δ$, and show that tangent bundle $TM$ can be coupled with the Lie algebra bundle $L$ if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with suc…
Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie's classification. The simplest type of lift is encoded by Lie algebra cohomology.
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。
On the homotopy invariance of the twisted Lie algebroid cohomology
本文证明了扭曲李代数胚同调的同伦不变性，并利用Poincaré引理推广和Mayer-Vietoris定理导出了Künneth公式，为计算传递李代数胚的扭曲同调提供了系统方法。
Transitive nilpotent Lie algebras of vector fields and their Tanaka prolongations
本文研究向量场可递幂零李代数的田中延拓导子来源问题，明确一般情形下0次导子由0次向量场给出、延拓恢复整个多项式向量代数，同时找出并刻画了特定伸缩情形下的奇异导子。
We study pseudo-Riemanniasn manifolds $(M,g)$ with transitive group of conformal transformation which is essential, i.e. does not preserves any metric conformal to $g$. All such manifolds of Lorentz signature with non exact isotropy representation of the stability subalgebra are described. A construction of essential conformally homogeneous manifolds with exact isotropy represenatation is given. Using spinor formalism, we prove that it provides all 4-dimensional non conformally flat Lorentzian 4-dimensional manifolds with transitive essentially conformal group.
Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine…
Transitive Courant algebroids and double symplectic groupoids
该研究将双重辛群胚的Lu-Weinstein构造推广至传递Courant代数胚且Atiyah代数胚可积的李双代数胚，分类了李群胚上的正合扭曲Courant代数胚，证明了扭曲CA-群胚基上的叶状结构，推广了现有双重辛群胚例子。
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