Question on Lie group regular actions

[mnb96]

Hello,

it is known that "Every regular G-action is isomorphic to the action of G on G given by left multiplication".
Is this true also when G is a Lie group?

There is an ambiguous sentence in Wikipedia that is confusing me. It says: "The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.". This sentence probably refers to the above statement about the isomorphism of regular actions and the action of G on itself, but I don't understand why it is supposed to be true for Lie group actions.

 

[Stephen Tashi]

. It says: "The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.".

I'm not an expert on this subject, but I think what happens is that the definition of "Isomorphism" changes when you move from groups to topological groups. The definitions of homomorphisms and isomorphism for a group don't have any requirements about the continuity of the mappings. So the fact that you proved something about a homomorphism for groups doesn't get you a proof of the same theorem for topological groups.

This sentence probably refers to the above statement about the isomorphism of regular actions and the action of G on itself, but I don't understand why it is supposed to be true for Lie group actions.

It's hard to interpret your sentence! When you say "it is supposed to be true for Lie Group actions", what does that mean?

 

[mnb96]

Hi Stephen!

thanks for your reply. You are right! I didn't think that in the context of Lie groups we have to change the definition of isomorphism and impose some constraints of continuity on the mapping.
This basically answers my question.