# Insights A Journey to The Manifold - Part I - Comments

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1. Oct 16, 2017

### Staff: Mentor

Last edited: Oct 16, 2017
2. Oct 19, 2017

### Greg Bernhardt

3. Oct 28, 2017

### Zafa Pi

4. Oct 28, 2017

### Staff: Mentor

This was a sloppy abbreviation. The group is defined on the open unit disc and the inversion on the open half of it. I combined both, because I didn't want to go through the entire definition and verification of a local Lie group, because this was not the point there. I primarily wanted to give an example which is not a global matrix group and which has a somehow unusual multiplication. I therefore quoted the source of the example for details. But as you ask, here is the actual definition of a local Lie group.

An $n-$parameter local Lie group consists of connected open subsets $\{0\} \in U_0 \subseteq U \subseteq \mathbb{R}^n$, a smooth group multiplication $U \times U \longrightarrow \mathbb{R}^n$ and a smooth inversion $U_0 \longrightarrow U$ with $0$ as identity element and the usual group axioms. The locality is given by the fact that the group operations only need to apply on a local area around the identity element. The same holds for the group axioms: they only have to hold where they are defined. This makes it different from a global Lie group, where those operations need to be defined everywhere.

But you're right, this has been a bit sloppy, since I left the details of the definition to the reader. (On my list of changes for an update. I have to see first where it can be done without taking too much space.)