Projective reparametrization of homogeneous curves

Boris Doubrov

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 1, page 129-133
  • ISSN: 0044-8753

Abstract

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We study the conditions when locally homogeneous curves in homogeneous spaces admit a natural projective parameter. In particular, we prove that this is always the case for trajectories of homogeneous nilpotent elements in parabolic spaces. On algebraic level this corresponds to the generalization of Morozov–Jacobson theorem to graded semisimple Lie algebras.

How to cite

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Doubrov, Boris. "Projective reparametrization of homogeneous curves." Archivum Mathematicum 041.1 (2005): 129-133. <http://eudml.org/doc/249480>.

@article{Doubrov2005,
abstract = {We study the conditions when locally homogeneous curves in homogeneous spaces admit a natural projective parameter. In particular, we prove that this is always the case for trajectories of homogeneous nilpotent elements in parabolic spaces. On algebraic level this corresponds to the generalization of Morozov–Jacobson theorem to graded semisimple Lie algebras.},
author = {Doubrov, Boris},
journal = {Archivum Mathematicum},
keywords = {homogeneous submanifold; symmetry algebra; nilpotent elements; $sl_2$-tripple; homogeneous submanifold; symmetry algebra; nilpotent elements; -triple},
language = {eng},
number = {1},
pages = {129-133},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Projective reparametrization of homogeneous curves},
url = {http://eudml.org/doc/249480},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Doubrov, Boris
TI - Projective reparametrization of homogeneous curves
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 1
SP - 129
EP - 133
AB - We study the conditions when locally homogeneous curves in homogeneous spaces admit a natural projective parameter. In particular, we prove that this is always the case for trajectories of homogeneous nilpotent elements in parabolic spaces. On algebraic level this corresponds to the generalization of Morozov–Jacobson theorem to graded semisimple Lie algebras.
LA - eng
KW - homogeneous submanifold; symmetry algebra; nilpotent elements; $sl_2$-tripple; homogeneous submanifold; symmetry algebra; nilpotent elements; -triple
UR - http://eudml.org/doc/249480
ER -

References

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  1. Bourbaki N., Éléments de mathématique, Fasc. XXXVIII: Groupes et algèbres de Lie. Chap. VII: Sous-algèbres de Cartan, éléments réguliers. Chap. VIII: Algèbres de Lie semi-simples déployées, Actualités scientifiques et industrielles, 1364, Paris, Hermann 1975. (1975) Zbl0329.17002MR0453824
  2. Cap A., Slovák J., Žádník V., On distinguished curves in parabolic geometries, Transform. Groups 9 (2004), 143–166. Zbl1070.53021MR2056534
  3. Doubrov B., Komrakov B., Rabinovich M., Homogeneous surfaces in three-dimensional affine geometry, In: Geometry and topology of submanifolds, VIII, Singapore, World Scientific 1996, 168–178. (1996) MR1434565
  4. Doubrov B., Komrakov B., Classification of homogeneous submanifolds in homogeneous spaces, Lobachevskii Journal of Mathematics 3 (1999), 19–38. (1999) Zbl0964.53035MR1743130
  5. Eastwood M., Slovák J., Preferred parametrizations on homogeneous curves, arXiv: math.DG/0311456. 
  6. Hermann R., Sophus Lie’s 1880 transformation group paper, Math. Sci. Press Brookline 1975. (1975) Zbl0406.22006MR0460053
  7. Jacobson N., Lie algebras, Intersci. Tracts in Pure and Appl. Math. 10, New-York–London, John Wiley and Sons 1962. (1962) Zbl0121.27504MR0143793
  8. Lie S., Theorie der Transformationgruppen, Bd. 3, Leipzig, Teubner, 1893. 
  9. Vinberg E., Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Sel. Math. Sov. 6 (1987), 15–35. (1987) Zbl0612.17010

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