Classification and Decomposition of Quaternionic Projective Transformations

TL;DR

This paper studies the structure and classification of transformations in the quaternionic projective linear group, including reversibility, dynamical types, and decomposition into simple elements, extending real projective transformation classifications.

Contribution

It provides an algebraic characterization of dynamical types in $ ext{PSL}(3, ext{H})$ and introduces a decomposition of elements into conjugates of real special linear group elements.

Findings

Reversibility problem solved for $ ext{PSL}(3, ext{H})$

Classification of elements based on dynamical types

Decomposition of elements into simple conjugate components

Abstract

We consider the projective linear group $\mathrm{PSL}(3,\mathbb{H})$. We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of $\mathrm{PSL}(3,\mathbb{H})$. We further decompose elements of $\mathrm{SL}(3,\mathbb{H})$ as products of simple elements, where an element $g$ in $\mathrm{SL}(3,\mathbb{H})$ is called $\textit{simple}$ if it is conjugate to an element of $\mathrm{SL}(3,\mathbb{R})$. We have also revisited real projective transformations and following Goldman's ideas, have offered a complete classification for elements of $\mathrm{SL }(3,\mathbb{R})$.