Computation of the left invariant vector field for SO(3)

In summary, the conversation is about the topic of Lie groups and the operations of left multiplication and pushforward. The person has been studying this topic and has been using some notes from online sources. They have also worked out some calculations for the special orthogonal group SO(3) but are unsure about the appearance of sin and cos after matrix multiplication. They mention a code in a comment on a forum post that may be relevant, although it is for a different example.
  • #1
nigelscott
135
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I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward.

I have been looking at these notes:

https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example
https://math.stackexchange.com/ques...ate-a-vector-in-a-left-invariant-vector-field
https://math.stackexchange.com/questions/1764479/left-invariant-vector-field-on-so3

I have worked out the following for SO(3).

so.jpg


Is this correct? I am not sure about the appearance of the sin and cos after the matrix multiplication.
 

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1. What is the significance of computing the left invariant vector field for SO(3)?

The left invariant vector field for SO(3) is a fundamental tool in understanding the group structure of SO(3), which is the group of rotations in three-dimensional space. It allows for the representation of rotations as vector fields, making it easier to analyze and manipulate them mathematically.

2. How is the left invariant vector field for SO(3) computed?

The computation of the left invariant vector field for SO(3) involves finding a basis for the tangent space at the identity element of the group. This is typically done using the Lie algebra representation of the group, which involves matrix operations and differentiation.

3. What is the relationship between the left invariant vector field and the Lie algebra of SO(3)?

The left invariant vector field is closely related to the Lie algebra of SO(3), as it can be seen as the infinitesimal generators of the group. This means that the vector field captures the local behavior of the group elements near the identity element, providing insight into the group's structure and properties.

4. Can the left invariant vector field for SO(3) be applied to other groups?

Yes, the concept of a left invariant vector field can be applied to other Lie groups, not just SO(3). However, the specific computation may vary depending on the group's structure and properties. The left invariant vector field is a powerful tool in the study of Lie groups, and it has many applications in various fields of mathematics and science.

5. What are some practical applications of the left invariant vector field for SO(3)?

The left invariant vector field for SO(3) has applications in robotics, computer graphics, and physics, among others. It can be used to analyze the motion and behavior of rigid bodies and to solve differential equations involving rotations. It is also essential in understanding the dynamics of mechanical systems and in developing control algorithms for robotic systems.

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