The Ortogonal Representation of SU2

In summary, the rotation group SO3 can be mapped to the 2 sphere by sending a rotation matrix to its first column. The fibres of this map can be described as rotations that have the same first column, and differ only by a rotation in the plane orthogonal to this column. This map can be represented as phi: SO3 ---> S2, where SO3 is the set of special matrices with PPt=I and det P=1 and S2 is the 2 sphere.
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Homework Statement



The rotation group SO3 may be mapped to the 2 sphere by sending a rotation matrix to its first column. Describe the fibres of this map

Homework Equations



SO3 are the special matrices P such that PPt=I and they are 3x3 matrices with det P =1

The Attempt at a Solution



ok so we just want to describe the inverse image of an element. I know the image of the element is a matrix which corresponds to some rotation is sent to a 2-sphere. But i don't really know how to describe this formally or in the other direction.i know we have 3x3 matrices with det =1 st. ppt

so our map (phi):SO3 ---> S2

but don't know where to go from here.
 
  • #3
Think of them as what they are, rotations. If two rotations R1 and R2 have the same first column then they map (1,0,0)=e1 to the same vector, call it v (the first column of the matrix). They both map e2 and e3 to vectors orthogonal to v. They must differ only by a rotation in the plane orthogonal to v, right?
 

Related to The Ortogonal Representation of SU2

1. What is the Ortogonal Representation of SU2?

The Ortogonal Representation of SU2 is a mathematical concept used to represent the special unitary group SU2 in a way that preserves its properties under orthogonal transformations.

2. How is the Ortogonal Representation of SU2 different from other representations?

The Ortogonal Representation of SU2 is unique in that it is a unitary representation that is also invariant under orthogonal transformations, making it useful for studying physical systems that involve both unitary and orthogonal symmetries.

3. What are some applications of the Ortogonal Representation of SU2?

The Ortogonal Representation of SU2 has many applications in physics, particularly in quantum mechanics and particle physics. It is also used in other fields such as signal processing and computer graphics.

4. How is the Ortogonal Representation of SU2 related to the special unitary group SU2?

The Ortogonal Representation of SU2 is a specific type of representation of the special unitary group SU2, which is a mathematical group used to describe the symmetries of physical systems. The Ortogonal Representation preserves the unitary and orthogonal properties of the group.

5. What are the mathematical properties of the Ortogonal Representation of SU2?

The Ortogonal Representation of SU2 has several important mathematical properties, including being a unitary representation, preserving the group's structure under orthogonal transformations, and being a basis for the Lie algebra of SU2. It is also used in the theory of spinors and has connections to the theory of quaternions.

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