Results in Mathematics

, Volume 18, Issue 3–4, pp 355–364 | Cite as

Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

  • Abraham A. Ungar


Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.


Product Space Binary Operation Semidirect Product Lorentz Factor Thomas Precession 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Richard H. Bruck, A Survey of Binary Systems, 2nd ed., Springer-Verlag, New York, 1966.zbMATHGoogle Scholar
  2. [2]
    Herbert Goldstein, Classical Mechanics, 2nd edn, Addison-Wesley, Menlo-Park, California, 1980.zbMATHGoogle Scholar
  3. [3]
    Dan Kaiman, The axis of rotation: analysis, algebra, geometry, Math. Mag. 62 (1989) 248–252.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Helmut Karzel, Inzidenzgruppen I, lecture notes by I. Pieper and K. Sörensen, Univ. Hamburg 1965, pp. 123-135.Google Scholar
  5. [5]
    William Kerby, On Infinite Sharply Multiply Transitive Groups, Hamburger Mathematische Einelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974.Google Scholar
  6. [6]
    William Kerby and Heinrich Wefelscheid, The maximal subnear-field of a neardomain, J. Algebra, 28 (1974) 319–325.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Günter Kist, Theorie der verallgemeinerten kinematischen Räume, Results Math. 12 (1987) 325–347.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Jerrold E. Marsden, Elementary Classical Analysis, Section 1.2, Freeman, New York, 1974.zbMATHGoogle Scholar
  9. [9]
    James G. Simmonds, A Brief on Tensor Analysis, Springer, New York, 1982.zbMATHCrossRefGoogle Scholar
  10. [10]
    Llewellyn H. Thomas, The motion of the spinning electron, Nature 117 (1926) 514.CrossRefGoogle Scholar
  11. [11]
    Llewellyn H. Thomas, The kinematics of an electron with an axis, Philos. Mag. S. 7 (1927) 1–23.Google Scholar
  12. [12]
    Llewellyn H. Thomas, iRecollections of the discovery of the Thomas precessional frequency, AIP Conf. Proc. No. 95, High Energy Spin Physics (Brookhaven National Lab, ed. G.M. Bunce), 1982, pp. 4-12.Google Scholar
  13. [13]
    Jacques Tits, Généralisation des groupes projectifs, Acad. Roy. Belg. Cl. Sci. Mém. Coll. 5e Ser. 35 (1949) 197–208,224-233, 568-589,756-773.MathSciNetzbMATHGoogle Scholar
  14. [14]
    George E. Uhlenbeck, Fifty years of spin: personal reminiscences, Phys. Today, June (1976) 43-48.Google Scholar
  15. [15]
    Abraham A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1(1988)57–89.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Abraham A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, Results Math. 6 (1989) 168–179.MathSciNetGoogle Scholar
  17. [17]
    Abraham A. Ungar, Weakly associative group(oid)s, Results Math. 17 (1990) 149–168.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Abraham A. Ungar A unified approach for solving quadratic, cubic and quartic equations by radicals, Int. J. Comp. Math. Appl. 19 (1990) 33–39.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Abraham A. Ungar, Quasidirect product groups and the Lorentz transformation group, in T.M. Rassias (ed.), Constantin Caratheodory: An International Tribute, World Sci. Pub. NJ, 1991.Google Scholar
  20. [20]
    Heinz Wärding, Theorie der Fastkörper, Thaies Verlag, W. Germany, 1987.Google Scholar
  21. [21]
    Heinrich Wefelscheid, ZT-Subgroups of sharply 3-transitive Groups, Proc. Edinburgh Math. Soc., 23(1980) 9–14.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Heinrich Wefelscheid, K-loops und die algebraische Struktur der zulässigen Geschwindigkeiten in der speziellen Relativitätstheorie, preprint.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1990

Authors and Affiliations

  • Abraham A. Ungar
    • 1
  1. 1.Department of MathematicsNorth Dakota State UniversityFargoUSA

Personalised recommendations