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Adjoint and Coadjoint Orbits of the Poincaré Group

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Abstract

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincaré group.

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References

  1. Arens, R.: Classical Lorentz invariant particles. J. Math. Phys. 12, 2415–2422 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bargmann, V., Wigner, E.P.: Group theoretical discussion of relativistic wave equations. Proc. Natl. Acad. Sci. USA 34, 211–223 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  3. Burgoyne, N., Cushman, R.: Conjugacy classes in linear groups. J. Algebra 44, 339–362 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, UK (1984)

    MATH  Google Scholar 

  5. Rawnsley, J.H.: Representations of a semi-direct product by quantization. Math. Proc. Camb. Philos. Soc. 78, 345–350 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  6. Souriau, J.-M.: Structure of Dynamical Systems. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  7. Wigner, E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute, University of Utrecht, 3508TA, Utrecht, The Netherlands

    Richard Cushman & Wilberd van der Kallen

Authors
  1. Richard Cushman
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  2. Wilberd van der Kallen
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Corresponding author

Correspondence to Wilberd van der Kallen.

Additional information

The author (R.C) was partially supported by European Community funding for the Research and Training Network MASIE (HPRN-CT-2000-00113).

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Cushman, R., van der Kallen, W. Adjoint and Coadjoint Orbits of the Poincaré Group. Acta Appl Math 90, 65–89 (2006). https://doi.org/10.1007/s10440-006-9031-8

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  • DOI: https://doi.org/10.1007/s10440-006-9031-8

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