Some extensions of quaternions and symmetries of simply connected space forms

TL;DR

This paper develops a unified algebraic framework for representing and decomposing various symmetry groups in 3 and 4 dimensions, extending classical quaternionic results to broader algebraic structures.

Contribution

It introduces a comprehensive approach that encompasses classical and new symmetry representations using extended quaternionic algebras.

Findings

Unified algebraic framework for symmetry groups in 3 and 4 dimensions.

New decomposition of group rotations in four dimensions.

Extension of classical quaternionic representations to split biquaternions.

Abstract

It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the groups of unit--norm elements in the algebras of real quaternions, biquaternions (complex quaternions) and dual quaternions, respectively. In this work, we present a unified framework that allows a wider scope on the subject and includes all the classical results related to the action in dimension 3 and 4 of unit--norm elements of the algebras described above and the algebra of split biquaternions as particular cases. We establish a decomposition of unit--norm elements in all cases and obtain as a byproduct a new decomposition of the group rotations in dimension 4.