Proving del_X(Y)=0.5[X,Y] in Lie Group Geometry

In summary, the conversation discusses the use of the Levi-Civita connection and bi-invariance to prove that del_X(Y) = 0.5 [X,Y] for left-invariant vector fields on a Lie group with a bi-invariant metric. This is achieved by using the formula for the Levi-Civita connection with the inner product.
  • #1
sroeyz
5
0
Hello,
I seem to be having difficulty proving something.
I hope you can help me.

I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X).

Let G be a lie group, with a bi-invariant metric , g , on G.
I want to prove that del_X(Y) = 0.5 [X,Y] (Lie brackets) , whenever X,Y are left-invariant vector fields on G.

Thanks in advance.
 
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  • #2
sroeyz said:
Hello,
I seem to be having difficulty proving something.
I hope you can help me.

I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X).

Let G be a lie group, with a bi-invariant metric , g , on G.
I want to prove that del_X(Y) = 0.5 [X,Y] (Lie brackets) , whenever X,Y are left-invariant vector fields on G.

Thanks in advance.


The Levi-Civita connection can be expressed via the inner product (which is a fundamental result in metric differential geometry). Use this formula and bi-invariance to obtain the result.
 

1. What is the significance of proving del_X(Y)=0.5[X,Y] in Lie Group Geometry?

The equation del_X(Y)=0.5[X,Y] is a fundamental result in Lie Group Geometry, which is a branch of mathematics that deals with groups of continuous transformations. It is important because it allows us to understand the behavior of vector fields on Lie groups, which have applications in various fields such as physics and engineering.

2. What is the Lie derivative and how is it related to del_X(Y)=0.5[X,Y]?

The Lie derivative is a way of measuring the change of a vector field along another vector field. It is denoted by del_X(Y) and can be calculated using the equation del_X(Y)=0.5[X,Y], where [X,Y] is the Lie bracket (or commutator) of the two vector fields X and Y.

3. How is the proof of del_X(Y)=0.5[X,Y] in Lie Group Geometry carried out?

The proof of del_X(Y)=0.5[X,Y] involves using the properties of Lie brackets and the structure of Lie groups. It is a complex proof that relies on advanced mathematical concepts such as differential geometry and group theory.

4. What are some real-world applications of del_X(Y)=0.5[X,Y] in Lie Group Geometry?

The equation del_X(Y)=0.5[X,Y] has many applications in physics, particularly in the study of symmetries and conservation laws. It is also used in engineering, specifically in fields such as control theory and robotics.

5. Are there any variations of the equation del_X(Y)=0.5[X,Y] in different areas of mathematics?

Yes, there are variations of this equation in other branches of mathematics, such as differential geometry and differential equations. These variations may have different forms, but they all share the same fundamental idea of measuring the change of a vector field along another vector field.

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