1Institute of Mathematics, University of the Philippines, Diliman, Philippines 1101 2Dept. of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U.S.A.
Journal of Lie Theory, Tome 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 $G/ (\pm I_2)\simeq O(4)$.
1
Institute of Mathematics, University of the Philippines, Diliman, Philippines 1101
2
Dept. of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U.S.A.
Clarisson Canlubo; Edgar Reyes. Reflections on S3 and Quaternionic Möbius Transformations. Journal of Lie Theory, Tome 22 (2012) no. 3, pp. 839-844. http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a10/
@article{JOLT_2012_22_3_a10,
author = {Clarisson Canlubo and Edgar 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},
url = {http://geodesic.mathdoc.fr/item/JOLT_2012_22_3_a10/}
}
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