nLab
proper Lie groupoid

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

A Lie groupoid (C 1tsC 0)(C_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} C_0) is proper if its underlying topological groupoid is a proper topological groupoid, hence if

(s,t):C 1C 0×C 0 (s,t) : C_1 \to C_0 \times C_0

is a proper map.

So in particular the automorphism group of any object in a proper Lie groupoid is a compact Lie group. In this sense proper Lie groupoids generalize compact Lie groups.

Examples

References

  • M.J. Pflaum, H. Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids (arXiv:1101.0180)

Revised on February 2, 2012 11:42:51 by Urs Schreiber (82.169.65.155)