# nLab local Lie group

group theory

### Cohomology and Extensions

#### Differential geometry

differential geometry

synthetic differential geometry

∞-Lie theory

# 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 infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

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