SO\((3,\mathbb{C})\) Representation and Action on a Homogeneous Space in \(\mathbb{C}^3\)

Emilija Celakoska; Vesna Celakoska-Jordanova; Dushan Chakmakov

doi:10.26713/cma.v9i4.874

Authors

Emilija Celakoska Department of Mathematics and Informatics, Faculty of Mechanical Engineering, SS. Cyril and Methodius University, Skopje
Vesna Celakoska-Jordanova Institute of Mathematics, Faculty of Natural Sciences and Mathematics, SS. Cyril and Methodius University, Skopje
Dushan Chakmakov Department of Mathematics and Informatics, Faculty of Mechanical Engineering, SS. Cyril and Methodius University, Skopje

DOI:

https://doi.org/10.26713/cma.v9i4.874

Keywords:

complex special orthogonal group, Boost, Polar decomposition, Group link

Abstract

We consider a homogeneous manifold \(\mathcal{H}\) embedded in \(\mathbb{C}^3\) composed of complex vectors with constraints, potentially representing space of complex velocities. The imposed constraints include orthogonality between the real and the imaginary parts of vectors which together with the non-conjugate scalar product provide real vector magnitudes. The corresponding representation of the group SO\((3,\mathbb{C})\) acting on \(\mathcal{H}\) which is in agreement to the polar decomposition of complex orthogonal matrices is also discussed.

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