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Parametric realization of the Lorentz transformation group in pseudo-Euclidean spaces. (English) Zbl 1352.20043

In this paper, the author gives the unique parametrization \(\Lambda=\Lambda(P, O_n, O_m)\) of a Lorentz transformation \(\Lambda \). Since a Lorentz transformation is a linear transformation leaving in the pseudo-Euclidean space \(\mathbb R^{m,n}\) the pseudo-Euclidean inner product invariant, in the above parametrization \(P \in \mathbb R^{m,n}\) is a real \(n \times m\) matrix, \(O_n \in SO(n)\) and \(O_m \in SO(m)\) are special ortogonal matrices taking \(P\) into \(O_n P\) and \(P O_m\), respectively. Using this parametrization, the author expresses the Lorentz transformation composition law in terms of parameter composition. The parameters of \(\Lambda\) \(O_n\) and \(O_m\) are called a left rotation and a right rotation, respectively, of \(P \in \mathbb R^{m,n}\). The left as well as the right rotations form a group. The pair \((O_n, O_m)\) is called a bi-rotation. The parameter \(P\) does not form a group. It forms a non-associative generalization of the commutative group, it is called bi-gyrocommutative bi-gyrogroup. The abstract gyrogroups (the case \(m=1\), \(n>1\)) were introduced and studied in [A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group”, Found. Phys. Lett. 1, No. 1, 57–89 (1988; doi:10.1007/BF00661317)]. In this paper, the abstract bi-gyrogroups are extensively studied. Similar as gyrogroups, bi-gyrogroups can play a universal computational role that extends far beyond the domain of Lorentz transformations in pseudo-Euclidean spaces.

MSC:

20N02 Sets with a single binary operation (groupoids)
20N05 Loops, quasigroups
15A63 Quadratic and bilinear forms, inner products
22E43 Structure and representation of the Lorentz group
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