M\"obius transformation for left-derivative quaternion holomorphic functions

TL;DR

This paper explores the structure and properties of quaternionic holomorphic transformations, extending the classical M"obius transformations to quaternion functions, and investigates their algebraic and geometric features with potential physical applications.

Contribution

It introduces a general group encompassing quaternionic holomorphic transformations, analyzes its algebraic structure, and connects it to known groups like the M"obius and Heisenberg groups, with matrix representations and physical implications.

Findings

The group of quaternionic holomorphic transformations includes rotations, dilations, and translations.

The Lie algebra of the group is neither simple nor semi-simple.

QHT has a unique fixed point at infinity and does not admit a cross-ratio.

Abstract

Holomorphic quaternion functions only admit affine functions; thus, the M\"obius 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 M\"obius 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.