Haar Measure on SO(4): Exploring References

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In summary, Haar Measure on SO(4) is a mathematical concept that refers to a probability measure on the special orthogonal group of degree 4. This measure is used to explore the structure and properties of this group, and is often studied through various references and research methods such as the Peter-Weyl theorem and the Plancherel formula. It has applications in fields such as physics and engineering, and continues to be a topic of interest in mathematical research.
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Bobhawke
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I wasn't quite sure where to post this question, so please forgive me if I chose the wrong place.

Essentially I'm looking for an explicit expression for the Haar measure on SO(4), i.e. in terms of angles, or if you prefer, expressed in terms of the Lebesgue integral over a subset of the Lie algebra.

Does anyone know any references which might contain this?

Thank you.
 
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  • #2
Bobhawke said:
I wasn't quite sure where to post this question, so please forgive me if I chose the wrong place.

Essentially I'm looking for an explicit expression for the Haar measure on SO(4), i.e. in terms of angles, or if you prefer, expressed in terms of the Lebesgue integral over a subset of the Lie algebra.

Does anyone know any references which might contain this?

Thank you.

There's one in the appendix of http://arxiv.org/abs/0805.0267. There is at least one typo there, but the overall construction is using the correct methods.
 
  • #3
Thank you!
 

1. What is Haar Measure on SO(4)?

Haar Measure on SO(4) is a mathematical concept that describes a way of measuring the "size" of elements in the special orthogonal group of dimension 4. It is a tool that is used in various areas of mathematics, such as group theory and harmonic analysis.

2. How is Haar Measure on SO(4) defined?

Haar Measure on SO(4) is defined as a non-negative function that assigns a "measure" or "volume" to each element in the special orthogonal group of dimension 4. This measure is invariant under left and right translations, meaning it does not change when an element is multiplied on the left or right by another element in the group.

3. Why is Haar Measure on SO(4) important?

Haar Measure on SO(4) is important because it allows for the integration of functions over the special orthogonal group of dimension 4. This is useful in many areas of mathematics, such as in probability theory and representation theory.

4. How is Haar Measure on SO(4) calculated?

Haar Measure on SO(4) is typically calculated using integration techniques. It involves finding the "volume" of a particular element in the group and then integrating over all elements in the group. This can be a complex process and often requires advanced mathematical techniques.

5. What are some applications of Haar Measure on SO(4)?

Haar Measure on SO(4) has many applications in mathematics, including in the study of Lie groups, Fourier analysis, and probability theory. It is also used in physics and engineering, particularly in the fields of signal processing and quantum mechanics.

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