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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.

51A25 Algebraization in linear incidence geometry
20N05 Loops, quasigroups
51A45 Incidence structures embeddable into projective geometries
83A05 Special relativity
Full Text: DOI
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