Journal of Lie Theory

no. 3
Reflections on S3 and Quaternionic Möbius Transformations
Clarisson Canlubo1; Edgar Reyes2
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, 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

Clarisson Canlubo  1 ; Edgar Reyes  2

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, Volume 22 (2012) no. 3, pp. 839-844. doi: 10.5802/jolt.694
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     title = {Reflections on {S\protect\textsuperscript{3}} and {Quaternionic} {M\"obius} {Transformations}},
     journal = {Journal of Lie Theory},
     pages = {839--844},
     year = {2012},
     volume = {22},
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     doi = {10.5802/jolt.694},
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