Algebraic characterization of the isometries of the hyperbolic 5-space

TL;DR

This paper provides an algebraic framework for classifying and understanding the isometries of hyperbolic 5-space using quaternionic matrix representations, including conjugacy and centralizer classes.

Contribution

It introduces an algebraic characterization of isometries in hyperbolic 5-space via quaternionic matrices, detailing their dynamical types and conjugacy classes.

Findings

Classification of isometries into dynamical types

Determination of conjugacy classes in GL(2, H)

Analysis of centralizers and z-classes

Abstract

Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in $GL(2, \H)$.