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arXiv · 1906.11370

Some extensions of quaternions and symmetries of simply connected space forms

Gerardo Arizmendi·Marco Antonio Pérez-de la Rosa
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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.

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Gerardo Arizmendi, Marco Antonio Pérez-de la Rosa. 2019-06-26. Some extensions of quaternions and symmetries of simply connected space forms. https://arxiv.org/abs/1906.11370

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