Regular vs. classical M\"obius transformations of the quaternionic unit ball

TL;DR

This paper explores the properties of regular M"obius transformations of the quaternionic unit ball, comparing them with classical transformations and examining their relation to the Poincaré metric, including a quaternionic Schwarz-Pick lemma analog.

Contribution

It introduces and analyzes regular M"obius transformations in the quaternionic setting, highlighting their differences from classical transformations and their geometric properties.

Findings

Regular M"obius transformations are included in slice regular functions.

Comparison between regular and classical M"obius transformations of the quaternionic ball.

A quaternionic analog of the Schwarz-Pick lemma is established.

Abstract

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular M\"obius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular M\"obius transformations and the Poincar\'e metric of B, which is preserved by the classical M\"obius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.