nLab
local Lie group

Context

Group Theory

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

\infty-Lie theory

∞-Lie theory

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

Contents

Idea

A local Lie group (rarely also called Lie groupuscule) is a local / stalkwise version of a Lie group, containing information about the product operation in arbitrary small open neighborhoods of the unit element; the multiplication is defined only when the factors are sufficiently close to the unit element, and if the consecutive products of triples in both ways are defined they are associative. To every local Lie group one functorially associates its Lie algebra.

Every real Lie algebra is a Lie algebra of some local Lie group. Or in more modern and precise phrasing, the category of real local Lie groups is equivalent to the category of finite-dimensional real Lie algebras. This has been proved by Sophus Lie as his famous third theorem. The extension to the global Lie theory has been possible only after works of Élie Cartan, who extended the equivalence to the equivalence between the category of real Lie algebras and connected simply connected Lie groups.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on February 7, 2013 14:12:56 by Urs Schreiber (82.113.121.156)