# Question on Lie group regular actions

• mnb96
In summary, the conversation discusses the validity of the statement that "Every regular G-action is isomorphic to the action of G on G given by left multiplication" in the context of Lie group actions. It is mentioned that this statement may no longer hold for continuous group actions, which may be due to a change in the definition of isomorphism for topological groups. The speaker also expresses confusion about the supposed truth of this statement for Lie group 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 It says: "The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.".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.

mnb96 said:
. It says: "The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.".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?

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.

## 1. What is a Lie group regular action?

A Lie group regular action is a type of transformation that is defined by a Lie group on a set. This means that the transformation is smooth and preserves the structure of the set, which is often a manifold. In simpler terms, it is a continuous and smooth operation on a space that preserves its geometric properties.

## 2. How is a Lie group regular action different from other types of actions?

A Lie group regular action is different from other types of actions because it is defined by a Lie group, which is a type of mathematical object that is used to study symmetries. This means that the transformation is not arbitrary, but is instead based on a specific structure that has properties that can be analyzed and studied.

## 3. What are some examples of Lie group regular actions?

Some examples of Lie group regular actions include rotations in three-dimensional space, translations in Euclidean space, and reflections in a plane. These actions are defined by Lie groups such as the special orthogonal group SO(3) for rotations and the affine group Aff(3) for translations.

## 4. What is the importance of studying Lie group regular actions?

Studying Lie group regular actions is important because they play a crucial role in many areas of mathematics and physics. They are used to study symmetries in geometric structures, such as manifolds, and are also essential in understanding the behavior of physical systems described by symmetry groups.

## 5. How are Lie group regular actions used in practical applications?

Lie group regular actions have a wide range of practical applications. They are used in physics to describe the symmetries of physical systems, in robotics to model the movements of robots, and in computer graphics to represent and manipulate 3D objects. They are also used in optimization problems, such as finding the shortest path between two points on a curved surface.

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