Communications in Mathematical Physics

, Volume 169, Issue 2, pp 385–395 | Cite as

The galilean group in 2+1 space-times and its central extension

  • S. K. Bose


The problem of constructing the central extensions, by the circle group, of the group of Galilean transformations in two spatial dimensions; as well as that of its universal covering group, is solved. Also solved is the problem of the central extension of the corresponding Lie algebra. We find that the Lie algebra has a three parameter family of central extensions, as does the simply-connected group corresponding to the Lie algebra. The Galilean group itself has a two parameter family of central extensions. A corollary of our result is the impossibility of the appearance of non-integer-valued angular momentum for systems possessing Galilean invariance.


Neural Network Statistical Physic Angular Momentum Complex System Nonlinear Dynamics 
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  1. 1.
    Wigner, E.P.: Ann. of Math.40, 149–204 (1939)MathSciNetGoogle Scholar
  2. 2.
    Mukunda, N., Sudarshan, E.C.G.: Classical Dynamics: A Modern Perspective, New York: John Wiley & Sons, 1974, Chapter 19Google Scholar
  3. 3.
    Bargmann, V.: Ann. of Math.59, 1–46 (1954)Google Scholar
  4. 4.
    Levy-Leblond, J.: J. Math. Phys.4, 776–788 (1963)CrossRefGoogle Scholar
  5. 5.
    Borchers, H.J., Sen, R.N.: Commun. Math. Phys.42, 101–126 (1975)CrossRefGoogle Scholar
  6. 6.
    Wilczek, F.: Phys. Rev. Lett.49, 957–959 (1983)CrossRefGoogle Scholar
  7. 7.
    Divakaran, P.P.: Rev. Math. Phys.6, 167–205 (1994)CrossRefGoogle Scholar
  8. 8.
    Raghunathan, M.S.: Rev. Math. Phys.6, 207–225 (1994)CrossRefGoogle Scholar
  9. 9.
    Moore, C.C.: Trans. Am. Math. Soc.113, 40–63 (1964)Google Scholar
  10. 10.
    Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry — Methods and Applications, Part II. The Geometry and Topology of Manifolds, Berlin, Heidelberg, New York: Springer-Verlag, 1993, p. 29Google Scholar
  11. 11.
    Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1982, Chap. IVGoogle Scholar
  12. 12.
    Kirillov, A.A.: Elements of the Theory of Representations. Berlin, Heidelberg, New York: Springer-Verlag, 1976, pp. 18–22 and 222–224Google Scholar
  13. 13.
    Mackey, G.W.: The Theory of Unitary Group Representations, Chicago and London: The University of Chicago Press, 1976, pp. 199–202Google Scholar
  14. 14.
    Varadarajan, V.S.: Geometry of Quantum Theory, Second Edition, Berlin, Heidelberg, New York: Springer, 1985, Theorem 7.31, p. 268Google Scholar
  15. 15.
    Wilczek, F., Zee, A.: Phys. Rev. Lett.51, 2250–2252 (1983)CrossRefGoogle Scholar
  16. 16.
    Fröhlich, J., Marchetti, P.A.: Commun. Math. Phys.121, 177–223 (1989)CrossRefGoogle Scholar
  17. 17.
    Leinaas, J.M., Myrheim, J.: Nuovo Cimento37B, 1–23 (1977)Google Scholar
  18. 18.
    Goldin, G.A., Menikhoff, R., Sharp, D.H.: J. Math. Phys.22, 1664–1668 (1981)CrossRefGoogle Scholar
  19. 19.
    Imbo, T.D., Imbo, C.S., Sudarshan, E.C.G.: Phys. Lett.B235, 103–107 (1990)CrossRefGoogle Scholar
  20. 20.
    Fröhlich, J., Marchetti, P.A.: Nucl. Phys.B356, 533–573 (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • S. K. Bose
    • 1
  1. 1.Department of PhysicsUniversity of Notre DameNotre DameUSA

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