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Quaternions and Möbius transformations in dimension 4
 Kosina, Jan ; Lávička, Roman (advisor) ; Krump, Lukáš (referee)
In this work we describe transformations of the 3-dimensional and the 4- dimensional Euclidean space. First we show how one can elegantly describe re- flections and rotations in these dimensions using quaternions and we prove 2 structural theorems concerning the connection between the group of unit qua- ternions and the special orthogonal groups SO(3) and SO(4). Next we recall a part of the conformal mapping theory, which we use later in the description of the Möbius transformations. We define the Möbius transformations in dimension 4 as compositions of an even number of spherical inversions and reflections. We show that one can describe them also in dimension 4 as linear fractional trans- formations in an analogous way as in dimension 2, if we use quaternions instead of complex numbers. We then outline a classification of Möbius transformations into elliptic, loxodromic and parabolic classes and in dimension 4, we describe what each class looks like. 1
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