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2015 Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces
Abraham A. Ungar
J. Geom. Symmetry Phys. 38: 39-108 (2015). DOI: 10.7546/jgsp-38-2015-39-108

Abstract

The Lorentz transformation group ${\rm SO}(m,n)$, $m,n\in \mathbb{N}$, is a group of Lorentz transformations of order $(m,n)$, that is, a group of special linear transformations in a pseudo-Euclidean space $\mathbb{R}^{m,n}$ of signature $(m,n)$ that leave the pseudo-Euclidean inner product invariant. A parametrization of ${\rm SO}(m,n)$ is presented, giving rise to the composition law of Lorentz transformations of order $(m,n)$ in terms of parameter composition. The parameter composition, in turn, gives rise to a novel group-like structure that $\mathbb{R}^{m,n}$ possesses, called a bi-gyrogroup. Bi-gyrogroups form a natural generalization of gyrogroups where the latter form a natural generalization of groups. Like the abstract gyrogroup, the abstract bi-gyrogroup can play a universal computational role which extends far beyond the domain of pseudo-Euclidean spaces.

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Abraham A. Ungar. "Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces." J. Geom. Symmetry Phys. 38 39 - 108, 2015. https://doi.org/10.7546/jgsp-38-2015-39-108

Information

Published: 2015
First available in Project Euclid: 27 May 2017

zbMATH: 1352.20043
MathSciNet: MR3380218
Digital Object Identifier: 10.7546/jgsp-38-2015-39-108

Rights: Copyright © 2015 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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