Composition of elements of SO(4)

In summary, the conversation discusses the use of Givens' rotations in obtaining the whole group of ##SO(4)##. The question is whether every element of ##SO(4)## can be written as a product of a certain number of Givens' rotations, and why this is important. The conversation also mentions an expression of ##SO(4)## elements in terms of pairs of unit quaternions and their associated rotation matrices. It is suggested that these matrices are similar to those of ##SU(2)##.
  • #1
jk22
729
24
It's a beginner's question : if I consider compositions of Givens' rotations, how many should I combine to obtain the whole group, since those don't commute and hence their order is important ? Is it 6 ?
 
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  • #2
Can you be a bit more precise? What are "Givens' rotations"? Do you want to know whether every element of ##SO(4)## can be written as a product of certain elements and of certain length? And most of all: why?
 
  • #3
There exists an expression of SO(4) elements in terms of pairs of unit quaternions. Each quaternion has a SO(4) rotation matrix associated with it, and the two quaternions' rotation matrices are multiplied together. Those matrices are linear functions of their quaternions, much like SU(2) matrices.

Was that what jk22 was asking about?
 

1. What is the composition of elements in SO(4)?

The composition of elements in SO(4) refers to the types and quantities of atoms present in a molecule of SO(4). Specifically, SO(4) is composed of one sulfur atom, four oxygen atoms, and no other elements.

2. How are the elements in SO(4) arranged?

The elements in SO(4) are arranged in a specific geometric structure known as a tetrahedral shape. This means that the sulfur atom is at the center, with the four oxygen atoms bonded to it at equal distances, forming a pyramid-like shape.

3. What are the properties of SO(4)?

SO(4) is a colorless, odorless, and highly reactive molecule. It is soluble in water and forms strong acids when combined with water. It is also a strong oxidizing agent, meaning it can easily accept electrons from other molecules.

4. How is SO(4) formed?

SO(4) is formed through the combination of sulfur dioxide (SO2) and oxygen (O2) in the presence of a catalyst, such as vanadium pentoxide. This reaction results in the formation of sulfur trioxide (SO3), which then combines with water to form SO(4).

5. What are the common uses of SO(4)?

SO(4) has many industrial uses, such as in the production of sulfuric acid, which is used in the manufacturing of fertilizers, detergents, and batteries. It is also used in the production of paper, textiles, and other chemicals. In addition, SO(4) is an important component in the Earth's atmosphere and plays a role in the formation of acid rain.

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