Regular Moebius transformations of the space of quaternions

TL;DR

This paper explores the properties of regular quaternionic functions, establishing analogs to classical complex analysis results, and characterizes regular Möbius transformations as the bijections of the quaternionic unit ball.

Contribution

It introduces regular fractional and Möbius transformations in quaternionic analysis, extending classical concepts and providing a comprehensive characterization of these transformations.

Findings

All regular injective functions on H are affine.

The group of biregular functions on H is the affine group.

Regular Möbius transformations map the quaternionic unit ball onto itself.

Abstract

Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the quaternionic setting introduces new phenomena. This paper studies regular quaternionic transformations. We first find a quaternionic analog to the Casorati-Weierstrass theorem and prove that all regular injective functions from H to itself are affine. In particular, the group Aut(H) of biregular functions on H coincides with the group of regular affine transformations. Inspired by the classical quaternionic linear fractional transformations, we define the regular fractional transformations. We then show that each regular injective function from the Alexandroff compactification of H to itself is a regular fractional transformation. Finally, we study regular Moebius transformations, which map the unit ball B onto itself. All regular bijections from B to itself prove to be regular Moebius transformations.