Is there such an identity about SO(3)?

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In summary, T^i K_{ij} = K T^j K^{-1} is a formula that shows the relationship between the generators (Lie algebra elements) of SO(3) and the elements of SO(3) itself. This formula is valid for all Lie groups and involves repeated indices which imply summation. However, it should be noted that SO(3) is uncountable and cannot have a finite set of generators, contrary to what some may believe. This formula has been confirmed to be true and can also be applied to other Lie groups.
  • #1
kakarukeys
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[tex]T^i K_{ij} = K T^j K^{-1}[/tex]
repeated indices imply summation.
[tex]T^i[/tex] are the generators (Lie algebra elements) of SO(3).
i.e.[tex]T^i_{jk} = - \epsilon_{ijk}[/tex]
[tex]T^i \in so(3)[/tex]
[tex]K \in SO(3)[/tex]

How to show it's true?
Is there a universal formula for all Lie group?
 
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  • #2
Those elements do not generate SO(3). SO(3) is uncountable, it cannot have a finte set of generators.
 
  • #3
matt grime said:
Those elements do not generate SO(3). SO(3) is uncountable, it cannot have a finte set of generators.

For physicists, generators of a Lie group are elements of the corresponding Lie algebra, or maybe such elements multiplied by i. I don't know what the K's are.
 
  • #4
Yes. I have just clarified!
I also checked a special case, and it's true.
I don't know how to prove this identity besides brute-force calculation which gives no insight at all.
 

Related to Is there such an identity about SO(3)?

1. What is SO(3)?

SO(3) stands for the Special Orthogonal Group of three-dimensional Euclidean space. It is a mathematical group that represents the rotations of a three-dimensional object around a fixed point.

2. Why is SO(3) important?

SO(3) is important because it has many applications in physics and engineering, particularly in the study of rotational motion and symmetry. It is also used in computer graphics, robotics, and computer vision.

3. How is SO(3) related to other mathematical concepts?

SO(3) is a subgroup of the Special Orthogonal Group SO(n), which represents the rotations in n-dimensional space. It is also closely related to the Lie group SO(3,1), which is used in the theory of relativity.

4. Is there a unique identity element in SO(3)?

Yes, the identity element in SO(3) is the 3x3 identity matrix, which represents no rotation or a rotation of 0 degrees around all three axes.

5. How is SO(3) used in real-world applications?

SO(3) is used in various real-world applications, such as robotics, navigation systems, and computer graphics. It is also used in the study of molecules and their symmetry, and in the analysis of the Earth's rotation and movements.

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