Geometry of the slice regular M\"obius transformations of the quaternionic unit ball

TL;DR

This paper studies the geometric structure of slice regular Möbius transformations of the quaternionic unit ball, establishing a smooth manifold structure and relating it to Lie groups and principal fiber bundles.

Contribution

It introduces a smooth manifold structure on the set of quaternionic slice regular Möbius transformations and relates it to Lie groups and fiber bundle quotients.

Findings

Manifold structure on al() is established

Relation of al() to ig(1,1) Lie group is shown

The manifold is diffeomorphic to b  4 1  S^3

Abstract

For the quaternionic unit ball $\mathbb{B}$, let us denote by $\mathcal{M}(\mathbb{B})$ the set of slice regular M\"obius transformations mapping $\mathbb{B}$ onto itself. We introduce a smooth manifold structure on $\mathcal{M}(\mathbb{B})$, for which the evaluation(-action) map of $\mathcal{M}(\mathbb{B})$ on $\mathbb{B}$ is smooth. The manifold structure considered on $\mathcal{M}(\mathbb{B})$ is obtained by realizing this set as a quotient of the Lie group $\mathrm{Sp}(1,1)$, Furthermore, it turns out that $\mathbb{B}$ is a quotient as well of both $\mathcal{M}(\mathbb{B})$ and $\mathrm{Sp}(1,1)$. These quotients are in the sense of principal fiber bundles. The manifold $\mathcal{M}(\mathbb{B})$ is diffeomorphic to $\mathbb{R}^4 \times S^3$.