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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