Journal of Lie Theory

no. 3
Reflections on S3 and Quaternionic Möbius Transformations
Journal of Lie Theory, Volume 22 (2012) no. 3, pp. 839-844
Let $S^3$ be the set of unit quaternions, let ${\cal H}$ be the algebra of quaternions, and let ${\cal H}^{\ast}$ be the space of pure quaternions. It is an elementary fact that $S^3$ and ${\cal H}^{\ast}\cup \{\infty\}$ are homeomorphic spaces by a stereographic projection. We show that a reflection in $S^3$ induces a linear fractional transformation on ${\cal H}^{\ast}\cup \{\infty\}$ that is defined by a matrix in a symplectic group $Sp(2)$. In addition, we identify the left eigenvalues of such a matrix, and show the subgroup $G$ generated by these matrices satisfies $G/ (\pm I_2)\simeq O(4)$.
DOI: 10.5802/jolt.694
Classification: 51B10, 15B33
Keywords: Moebius transformation, quaternion 
@article{JOLT_2012_22_3_a10,
     author = {C. Canlubo and E. Reyes},
     title = {Reflections on {S\protect\textsuperscript{3}} and {Quaternionic} {M\"obius} {Transformations}},
     journal = {Journal of Lie Theory},
     pages = {839--844},
     year = {2012},
     volume = {22},
     number = {3},
     doi = {10.5802/jolt.694},
     zbl = {1250.15002},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.694/}
}
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C. Canlubo; E. Reyes. Reflections on S3 and Quaternionic Möbius Transformations. Journal of Lie Theory, Volume 22 (2012) no. 3, pp. 839-844. doi: 10.5802/jolt.694

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