Reflections on S3 and Quaternionic Möbius Transformations

Key Points

• The aim is to investigate how reflections in the unit quaternion set S 3 create linear transformations in pure quaternions.
• Defined the relationship between unit quaternions and pure quaternions using stereographic projection.
• Analyzed reflections in S 3 and their effects on linear fractional transformations in H *.
• Identified left eigenvalues of matrices in the symplectic group Sp(2) and their subgroup properties.
• Demonstrated that reflections in S 3 induce specific linear fractional transformations in H *.
• Identified distinct left eigenvalues corresponding to matrices in Sp(2).
• Established that the subgroup G generated by these matrices satisfies G/(I 2 ) isomorphic to O(4).