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Journal of Lie Theory 22 (2012), No. 3, 839--844
Copyright Heldermann Verlag 2012

Reflections on S3 and Quaternionic Möbius Transformations

Clarisson Canlubo
Institute of Mathematics, University of the Philippines, Diliman, Philippines 1101

Edgar Reyes
Dept. of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U.S.A.


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 $G/ (\pm I_2)\simeq O(4)$.

Keywords: Moebius transformation, quaternion.

MSC: 51B10, 15B33

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