2019/06/26 by Gerardo Arizmendi, Arizmendi, Gerardo, Marco Antonio Pérez‐de la Rosa +1
Computer Science · Mathematics · #22E43 #30G35 #37E45 #51M10 #53A17 #70B10 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1906.11370
openalex publication_date 2019/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the groups of Euclidean rotations in dimension 3 (isometries of S2), 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.