Geodesic
Reflections on S3 and Quaternionic M�bius Transformations
Journal of Lie theory, Tome 22 (2012) no. 3, pp. 839-844.

Voir la notice de l'article provenant de la source Heldermann Verlag

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 $G/ (\pm I_2)\simeq O(4)$.
Classification : 51B10, 15B33
Mots-clés : Moebius transformation, quaternion 
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     author = {C. Canlubo and E. Reyes },
     title = {Reflections on {S\protect\textsuperscript{3}} and {Quaternionic} {M�bius} {Transformations}},
     journal = {Journal of Lie theory},
     pages = {839--844},
     publisher = {mathdoc},
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     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JLT_2012_22_3_JLT_2012_22_3_a10/}
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C. Canlubo; E. Reyes . Reflections on S3 and Quaternionic M�bius Transformations. Journal of Lie theory, Tome 22 (2012) no. 3, pp. 839-844. http://geodesic.mathdoc.fr/item/JLT_2012_22_3_JLT_2012_22_3_a10/