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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
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[1] Benz, W.:Geometrische Transformationen unter besonderer Berücksichtigung der Lorentztransformationen, BI-Wiss.-Verlag, Mannheim, Leipzig, Wien, Zürich, 1992. · Zbl 0754.51005
[2] Grundhöfer, T. and Salzmann, H.: Locally compact double loops and ternary fields, in O. Chein, Pflugfelder, H. O. and Smith, J. D. H.,Quasigroups and Loops, Theory and Applications, Heldermann Verlag, Berlin, 1990. · Zbl 0749.51016
[3] Karzel, H. and Konrad, A.: Raum-Zeit-Welt und hyperbolische Geometrie,Beiträge zur Geometrie und Algebra Nr.29 (1994) TUM-M 9412, TU München, 175p.
[4] Karzel, H. and Konrad, A.: Eigenschaften angeordneter Räume mit hyperbolischer Inzidenzstruktur,Beiträge zur Geometrie und Algebra Nr.28 (1994), TUM-M 9415, TU München, 27-36. · Zbl 0873.51008
[5] Karzel, H., Pianta, S. and Stanik, R.: Generalized Euclidean and elliptic geometries, their connections and automorphism groups,J. Geom. 48 (1993), 109-143. · Zbl 0797.51020 · doi:10.1007/BF01226804
[6] Karzel, H., Sörensen, K. and Windelberg, D.:Einführung in die Geometrie, Göttingen, 1973. · Zbl 0248.50001
[7] Karzel, H. and Wefelscheid, H.: Groups with an involutory antiautomorphism and K-loops; application to space-time-world and hyperbolic geometry I,Results Math. 23 (1993), 338-354. · Zbl 0788.20034
[8] Kolb, E. and Kreuzer, A.: Geometry of kinematic K-loops,Abh. Math. Sem. Univ. Hamburg, to appear. · Zbl 0852.20062
[9] Kreuzer, A. and Wefelscheid H.: On K-loops of finite order,Results Math. 25 (1994), 79-102. · Zbl 0803.20052
[10] Schröder, E. M.:Vorlesungen über Geometrie, Bd. I, BI-Wiss.-Verlag, Mannheim, Leipzig, Wien, Zürich, 1991.
[11] Sperner, E.: Ein gruppentheoretischer Beweis des Satzes von Desargues in der absoluten Axiomatik,Arch. Math. 5 (1954), 458-468. Nachgedruckt in: Karzel, H. and Sörensen, K.:Wandel von Begriffsbildungen in der Mathematik, Darmstadt, 1984, pp. 177-188. · Zbl 0057.12602 · doi:10.1007/BF01898391
[12] Ungar, A. A.: Group-like structure underlying the unit ball in real inner product spaces,Results Math. 18 (1990), 335-364. · Zbl 0718.20035
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