What Defines a Local Lie Group in A Journey to The Manifold - Part I?

[fresh_42]

fresh_42 submitted a new PF Insights post

A Journey to The Manifold - Part I

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Part 2 has arrived!
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

 

[Zafa Pi]

fresh_42 submitted a new PF Insights post

A Journey to The Manifold - Part I
View attachment 213148Continue reading the Original PF Insights Post.

I'm having trouble right from the get go. If someone says they have a group on a set U, that means if x and y ∈ U then x⋅y ∈ U. But with your example that is not true.

 

[fresh_42]

I'm having trouble right from the get go. If someone says they have a group on a set U, that means if x and y ∈ U then x⋅y ∈ U. But with your example that is not true.

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.)