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
crpcanlubo@math.upd.edu.ph
Edgar Reyes
Dept. of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U.S.A.
ereyes@selu.edu
[Abstract-pdf]
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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