Some questions on vector fields on Lie groups

In summary, the conversation is discussing a problem involving a Lie group G with unit e. The participants are trying to prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude that G has a basis of vector fields. They also discuss the concept of left invariant vector fields and their relationship to the map Lg. They are unsure of how De transforms (gv) and are seeking clarification on the problem.
  • #1
jacobrhcp
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0

Homework Statement



Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication.

a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields.
b) Prove for every v in TeG the is a unique vector field V on G that is left invariant (in the sense that the pullback of Lg on V is just V for all g in G) with Ve=v

for the original problem statement, see

https://www.math.uu.nl/people/looijeng/difftop06eng.pdf [Broken]

Questions about this

Lg (v) is just (gv), but I'm not sure how De transforms (gv). It is defined as the map that transforms a curve tangent to a point b to a curve tangent to e(b) on some other chart in G. If I understood this right, then De would send (gv) to the tangent vector at the origin in the direction of gv? - this last sentence of my own I did not fully understand.

if someone could help clarify the problem a bit, not so much as give a solution... that would be nice.
 
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  • #2
Homework Equations none really, can't think of any. The Attempt at a SolutionI don't know how to answer this one yet...
 

1. What are vector fields on Lie groups?

Vector fields on Lie groups are smooth functions that assign a tangent vector to each point on a Lie group. They are used to describe the infinitesimal motions and transformations of the group.

2. How are vector fields on Lie groups related to differential equations?

Vector fields on Lie groups can be used to model and solve differential equations on the group. This is because the group structure allows for a natural way to define derivatives and integrate vector fields.

3. Can vector fields on Lie groups be generalized to other mathematical structures?

Yes, vector fields on Lie groups can be generalized to other structures such as smooth manifolds and Lie algebras. This allows for the study of vector fields on a wider range of mathematical objects.

4. How are vector fields on Lie groups used in practical applications?

Vector fields on Lie groups have a wide range of applications in physics, engineering, and computer science. They are used to model physical systems, design control systems, and in the development of machine learning algorithms.

5. Are there any open problems or areas of research related to vector fields on Lie groups?

Yes, there are many open problems and ongoing research in the field of vector fields on Lie groups. Some current areas of interest include the study of invariant vector fields, the use of vector fields in geometric mechanics, and the application of vector fields to the study of dynamical systems.

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