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A geometric construction of the K-loop of a hyperbolic space. (English) Zbl 0845.51002
Let $${\mathcal H} = (P, {\mathfrak G}, \equiv, \zeta)$$ be a hyperbolic space of dimension 2 or 3 with point set $$P$$, line set $${\mathfrak G}$$, congruence relation $$\equiv$$ on pairs of points and betweenness relation $$\zeta$$ [cf. the first author and A. Konrad [‘Raum-Zeit und hyperbolische Geometrie’ (Beiträge zur Geometrie und Algebra 29, Technische Universität München TUM-M9412) (1994)] and the first author, K. Sörensen and D. Windelberg [‘Einführung in die Geometrie’ (Uni-Taschenbücher 184, Vandenhoeck, Göttingen) (1973; Zbl 0248.50001)] where the description of such spaces by matrices over commutative Euclidean fields is worked out in detail. The authors study the group $$\Gamma$$ of motions (i.e. automorphisms of $$(P, {\mathfrak G})$$ preserving $$\equiv)$$ of $${\mathcal H}$$. The point reflections in $$\Gamma$$ are identified with the points of $$P$$. The composition of different point reflections $$\widetilde a$$, $$\widetilde b$$ results in fixed point free mappings $$\beta = \widetilde a \circ \widetilde b$$ each of which determines a unique line $$L_\beta$$ (the join of $$a$$ and $$b)$$ and fixes all planes containing $$\L_\beta$$. Fixing a point $$0 \in P$$, the authors then get a set $$P^+ : = \{\widetilde o \circ \widetilde a |a \in P\}$$ of motions which operates sharply transitive on $$P$$ giving rise to a loop operation $$+$$ which in fact is shown to make $$(P,+)$$ into a $$K$$-loop $$(K$$-loops as defined in this paper coincide with Bol-loops that have the automorphic inverse property [cf. A. Kreuzer [‘Inner mappings of Bol loops’ (Beiträge zur Algebra und Geometrie 33, Technische Universität München TUM-M9509) (1995)]. The isomorphism between $$\Gamma$$ and the homogeneous orthochronous proper Lorentz group [cf. the first author and A. Konrad [loc. cit.]) maps the motions of the form $$\beta$$ onto the Lorentz boosts.
Using the $$K$$-loop structure on $$P$$ the authors finally show that the lines of $${\mathcal H}$$ can be recovered as sets of the form $$a + \{x \in P |x + b = b + x\}$$ where $$a,b \in P$$, $$b \neq 0$$. Thus, in particular, the set of relativistic admissible velocities becomes a 3-dimensional hyperbolic space.

##### MSC:
 51A25 Algebraization in linear incidence geometry 20N05 Loops, quasigroups 51A45 Incidence structures embeddable into projective geometries 83A05 Special relativity
##### Keywords:
Bruck loop; Lorentz group; $$K$$-loop; hyperbolic space
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##### References:
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