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Geometriae Dedicata

, Volume 58, Issue 3, pp 227–236 | Cite as

A geometric construction of the K-loop of a hyperbolic space

  • Helmut Karzel
  • Heinrich Wefelscheid
Article

Abstract

It is well known that the homogeneous orthochronous proper Lorentzgroup Γ is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost β ε Γ \ {id} there corresponds in the hyperbolic space exactly one lineL β such that β fixes each of the two ends ofL β . Furthermore β has no fixed points but each plane containingL β is fixed by β. If we fix a pointo, then to each other pointa there is exactly one boosta+ ε Γ such thatLa+ is the line joiningo anda anda+(o)=a. The set P of points of the hyperbolic space is turned in a K-loop (P, +) bya+b:=a+(b). Each line of the hyperbolic space has the representationa+Z(b) wherea, b εP,b ≠ 0 andZ(b):= {x εP |x+b=b+x}.

Mathematics Subject Classifications (1991)

51A25 20N05 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Helmut Karzel
    • 1
  • Heinrich Wefelscheid
    • 2
  1. 1.Mathematisches InstitutTechnische Universität MünchenMunichGermany
  2. 2.Fachbereich 11: MathematikUniversität DuisburgDuisburgGermany

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