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arXiv · 1508.01933

Möbius transformation for left-derivative quaternion holomorphic functions

Sergio Giardino
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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. They 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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Sergio Giardino. 2015-08-08. Möbius transformation for left-derivative quaternion holomorphic functions. https://doi.org/10.1007/s00006-016-0673-y

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