Isometries of the quaternionic hyperbolic line

TL;DR

This paper classifies the isometries of the quaternionic hyperbolic line by analyzing matrices in the unitary group U(1,1;H) through complex representations and characteristic polynomial computations.

Contribution

It provides a detailed classification of quaternionic hyperbolic isometries using complex matrix representations and polynomial analysis, a novel approach in this context.

Findings

Classification of matrices in U(1,1;H)

Use of complex representation phi(P)

Analysis of characteristic polynomial f

Abstract

We give a classification of the matrices in the unitary group U(1,1;H),where H is the division ring of the real quaternions. To this end, we consider the complex representation phi(P) for P in U(1,1;H). Next, we compute the characteristic polynomial f of the 4x4 complex matrix phi(P) and then study the sign of the resultant of f and its derivative f'.