Abstract
Holomorphic quaternion functions only admit affine functions; thus, the Möbius transformation for these functions, which we call quaternionic holomorphic transformation (QHT), only comprises similarity transformations. We determine a general group \({\mathsf{X}}\) which has the group \({\mathsf{G}}\) of QHT as a particular case. Furthermore, we observe that the Möbius group and the Heisenberg group may be obtained by making \({\mathsf{X}}\) more symmetric. We provide matrix representations for the group \({\mathsf{X}}\) and for its algebra \({\mathfrak{x}}\). The Lie algebra is neither simple nor semi-simple, and so it is not classified among the classical Lie algebras. We prove that the group \({\mathsf{G}}\) comprises \({\mathsf{SU}(2,\mathbb{C})}\) rotations, dilations and translations. The only fixed point of the QHT is located at infinity, and the QHT does not admit a cross-ratio. Physical applications are addressed at the conclusion.
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Giardino, S. Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions. Adv. Appl. Clifford Algebras 27, 1161–1173 (2017). https://doi.org/10.1007/s00006-016-0673-y
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DOI: https://doi.org/10.1007/s00006-016-0673-y