Related papers: A note on the global structure of proper Lie group…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert…

Given a Lie groupoid, we can form its orbit space, which carries a natural diffeology. More generally, we have a quotient functor from the Hilsum-Skandalis category of Lie groupoids to the category of diffeological spaces. We introduce the…

We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an…

We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $\pi_{k\geq n+1}$. The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We…

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups.…

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $\Gamma$-spaces. We show that equivariant principal $G$-bundles over split $\Gamma$-CW complexes $X$ can be effectively classified by means of…

We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures…

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups…

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the…

We show that the complete bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete…

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group…

We discuss two sorts of generalization of Lie groupoids. One is Lie $n$-groupoids defined as simplicial manifolds with trivial $\pi_{k\geq n+1}$. The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build…

In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung.…

Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $\alpha \colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the…

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and…

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to Lie algebras of dimension 7 such that the nilradical of them is 5-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ in Table…

Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal $G$-bundles, where $G$ is a complex reductive structure group. Flat connections on the affine line with a logarithmic…

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension~$7$, $8$ or $9$. They turn out to be closely related to…