Why is the tangent space of a lie group manifold at the origin the lie algebra?

In summary, differentiating a group element at the origin of the group manifold results in the generator, but differentiating a curve in the group manifold at the origin does not necessarily result in a generator. The bracket product of two left invariant vector fields is also left invariant, turning them into a Lie Algebra, and these vector fields are determined by their value at the identity.
  • #1
Bobhawke
144
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Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me.

I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this expression at the origin of the group manifold, ie the identity element, you are left with just the generator. So by differentiating a group element at the origin you get a generator. But this is not quite the same as differentiating a curve in the group manifold at the origin and getting a generator.
 
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  • #2
The bracket of two left invariant vector fields is itself left invariant. Thus the bracket product turns left invariant vector fields into a Lie Algebra. But left invariant vector fields are completely determined by their value at the identity.
 

1. What is a tangent space?

A tangent space is a mathematical concept that represents the set of all possible directions at a specific point on a smooth surface or manifold. It can be visualized as a vector space attached to a specific point on a curved surface, with each vector representing a possible direction of movement from that point.

2. What is a Lie group manifold?

A Lie group manifold is a type of mathematical space that combines the idea of a smooth surface or manifold with the concept of a group. It is a smooth, continuous space that also has a group structure, meaning it has a set of elements and operations that satisfy certain properties.

3. Why is the tangent space of a Lie group manifold important?

The tangent space of a Lie group manifold is important because it provides a way to study the local behavior of the manifold at a specific point. By examining the tangent space, we can understand the possible directions of movement from that point and how the manifold changes in those directions. This is particularly useful in the study of Lie groups, which often have curved and complex structures.

4. How is the tangent space of a Lie group manifold related to the Lie algebra?

The Lie algebra is a linear representation of the Lie group, meaning it is a vector space that captures the essential structure of the group. The tangent space of a Lie group manifold at the origin is isomorphic to the Lie algebra, meaning it has the same structure and properties. This allows us to use the linear structure of the Lie algebra to study the more complex structure of the Lie group.

5. Can the tangent space of a Lie group manifold at a point other than the origin be isomorphic to the Lie algebra?

Yes, the tangent space of a Lie group manifold at any point can be isomorphic to the Lie algebra, as long as that point is chosen carefully. The isomorphism between the tangent space and the Lie algebra is dependent on the specific point chosen, and the origin is often a convenient choice. However, in certain cases, a different point may provide a better understanding of the local behavior of the manifold.

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