Split octonions and generic rank two distributions in dimension five

Katja Sagerschnig

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 5, page 329-339
  • ISSN: 0044-8753

Abstract

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In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous space G ˜ 2 / P , where P is one of the maximal parabolic subgroups of the exceptional Lie group G ˜ 2 . In this article, we use the algebra of split octonions to give an explicit global description of the distribution corresponding to the homogeneous model.

How to cite

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Sagerschnig, Katja. "Split octonions and generic rank two distributions in dimension five." Archivum Mathematicum 042.5 (2006): 329-339. <http://eudml.org/doc/249791>.

@article{Sagerschnig2006,
abstract = {In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous space $\tilde\{G\}_2/P$, where $P$ is one of the maximal parabolic subgroups of the exceptional Lie group $\tilde\{G\}_2$. In this article, we use the algebra of split octonions to give an explicit global description of the distribution corresponding to the homogeneous model.},
author = {Sagerschnig, Katja},
journal = {Archivum Mathematicum},
keywords = {octonions; Cartan geometry; homogeneous model},
language = {eng},
number = {5},
pages = {329-339},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Split octonions and generic rank two distributions in dimension five},
url = {http://eudml.org/doc/249791},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Sagerschnig, Katja
TI - Split octonions and generic rank two distributions in dimension five
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 5
SP - 329
EP - 339
AB - In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous space $\tilde{G}_2/P$, where $P$ is one of the maximal parabolic subgroups of the exceptional Lie group $\tilde{G}_2$. In this article, we use the algebra of split octonions to give an explicit global description of the distribution corresponding to the homogeneous model.
LA - eng
KW - octonions; Cartan geometry; homogeneous model
UR - http://eudml.org/doc/249791
ER -

References

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  1. Baston R. J., Eastwood M. G., The Penrose transform: its interaction with representation theory, Oxford Science Publications, Clarendon Press, 1989. (1989) Zbl0726.58004MR1038279
  2. Čap A., Two constructions with parabolic geometries, Proceedings of the 25th Winter School on Geometry and Physics, Srní 2005, Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006), 11–38, preprint math.DG/0504389. Zbl1120.53013MR2287124
  3. Cartan E., Les systèmes de Pfaff à cinque variables et les équations aux dérivées partielles du seconde ordre, Ann. Sci. Ècole Normale Sup. 27 (1910), 109–192. (1910) MR1509120
  4. Sagerschnig K., Parabolic geometries determined by filtrations of the tangent bundle, to appear in Proceedings of the 25th Winter School on Geometry and Physics, Srni 2005, Rend. Circ. Mat. Palermo (2) Suppl. Zbl1114.53029MR2287136
  5. Springer T. A., Veldkamp F. D., Octonions, Jordan Algebras and Exceptional Groups, Springer, Berlin, 2000. Zbl1087.17001MR1763974
  6. Yamaguchi K., G 2 -geometry of overdetermined systems of second order, Analysis and Geometry in Several Complex Variables (Katata, 1997),Trends Math. (1999), 289–314, Trends Math. (1999). (1997) MR1699860

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