Question on group actions on vector spaces

In summary, the conversation discusses the conditions that a group G must satisfy in order for its orbit Gx to be a manifold for any given vector x in a vector space (e.g. \mathbb{R}^n). It is concluded that G must be a Lie group acting smoothly on R^n, and the multiplication must be continuous under the topology of the manifold. Additionally, G must act linearly on the vector space. This is necessary for the group structure to be preserved and for the orbit to be an immersed submanifold.
  • #1
mnb96
715
5
Hello,
If I am given a vector space (e.g. [itex]\mathbb{R}^n[/itex]), and a group [itex]G[/itex] that acts on [itex]\mathbb{R}^n[/itex], what are the conditions that [itex]G[/itex] must satisfy so that for any given [itex]x\in\mathbb{R}^n[/itex] its orbit [itex]Gx[/itex] is a manifold ?
 
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  • #2
If G is a Lie group acting smoothly on R^n, then the orbits will be immersed submanifolds.
 
  • #3
The orbit has the same group structure as G does by multiplication [tex]gx \cdot hx = (gh)x[/tex] and the obvious isomorphism [tex] g \mapsto gx[/tex]. So this means that your group G has to be a manifold. I'm guessing you'll want the multiplication to be continuous under the topology of the manifold or something, I'll have to think about itEDIT: Woops, not an isomorphism,. That's what happens when you use the word obvious.

Ok after thinking about this for a bit, I have to ask: did you perhaps want G to act linearly on the vector space?
 
Last edited:
  • #4
Office_Shredder said:
Ok after thinking about this for a bit, I have to ask: did you perhaps want G to act linearly on the vector space?

Uhm...Sorry, could you please specify what does "to act linearly" mean?
Thanks
 
  • #5
Every element of G is a linear map, so along with the normal group action rules you have

[tex]g \cdot (\alpha x + \beta y) = \alpha g \cdot x + \beta g \cdot y[/tex]

Otherwise the fact that we're living in a vector space is a happy coincidence and has very little bearing on the problem
 
  • #6
Ok. I got it, so yes...the group has to act linearly.
 

1. What is a group action on a vector space?

A group action on a vector space is a mathematical operation that maps elements of a group onto vectors in a vector space. This operation must satisfy certain properties, such as being distributive and associative, in order to be considered a valid group action.

2. How does a group action affect a vector space?

A group action on a vector space can transform the vectors within that space in a variety of ways. Depending on the specific group and its action, vectors may be rotated, reflected, translated, or otherwise transformed. This can provide insights into the symmetries and structures of the vector space.

3. What are some examples of group actions on vector spaces?

One example of a group action on a vector space is the rotation of a two-dimensional plane. The group of rotation operations can act on the vectors in the plane, moving them around in a circular motion. Another example is the reflection of a three-dimensional space, where the group of reflection operations can flip vectors across a specified plane.

4. How are group actions related to linear transformations?

Group actions and linear transformations have a close relationship, as both involve manipulating vectors in a vector space. In fact, every linear transformation can be viewed as a group action, and vice versa. This connection allows for the use of group theory in analyzing linear transformations.

5. What are the applications of group actions on vector spaces?

Group actions on vector spaces have many applications in mathematics and physics. They can be used to study symmetries and structures in geometry, to analyze the behavior of particles in physics, and to solve problems in abstract algebra. They also have practical applications in computer graphics and image processing.

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