The Ortogonal Representation of SU2

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SUMMARY

The discussion focuses on the mapping of the rotation group SO(3) to the 2-sphere (S²) through the first column of rotation matrices. It establishes that SO(3) consists of special orthogonal matrices P, satisfying PPT = I and det(P) = 1. The mapping function φ: SO(3) → S² is defined, and it is concluded that two rotations R₁ and R₂ with the same first column correspond to the same point on the 2-sphere, differing only by a rotation in the plane orthogonal to that column.

PREREQUISITES
  • Understanding of special orthogonal groups, specifically SO(3)
  • Familiarity with matrix operations and properties, particularly determinants
  • Knowledge of spherical geometry and the concept of the 2-sphere (S²)
  • Basic concepts of linear transformations and rotation matrices
NEXT STEPS
  • Study the properties and applications of SO(3) in 3D rotations
  • Explore the relationship between rotation matrices and quaternions
  • Learn about the fibers of mappings in differential geometry
  • Investigate the implications of the orthogonal representation in physics, particularly in quantum mechanics
USEFUL FOR

This discussion is beneficial for mathematicians, physicists, and computer scientists interested in rotational dynamics, geometric transformations, and the mathematical foundations of 3D graphics.

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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.
 
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?
 

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