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Insights A Journey to The Manifold - Part I - Comments

  1. Oct 16, 2017 #1

    fresh_42

    Staff: Mentor

    Last edited: Oct 16, 2017
  2. jcsd
  3. Oct 19, 2017 #2
  4. Oct 28, 2017 #3

    Zafa Pi

    User Avatar
    Gold Member

  5. Oct 28, 2017 #4

    fresh_42

    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.)
     
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