| Title:
|
Split octonions and generic rank two distributions in dimension five (English) |
| Author:
|
Sagerschnig, Katja |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
5 |
| Year:
|
2006 |
| Pages:
|
329-339 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| MSC:
|
53C30 |
| MSC:
|
58A30 |
| idZBL:
|
Zbl 1164.53362 |
| idMR:
|
MR2322419 |
| . |
| Date available:
|
2008-06-06T22:50:06Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108039 |
| . |
| Reference:
|
[1] Baston R. J., Eastwood M. G.: The Penrose transform: its interaction with representation theory.Oxford Science Publications, Clarendon Press, 1989. Zbl 0726.58004, MR 1038279 |
| Reference:
|
[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. Zbl 1120.53013, MR 2287124 |
| Reference:
|
[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. MR 1509120 |
| Reference:
|
[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. Zbl 1114.53029, MR 2287136 |
| Reference:
|
[5] Springer T. A., Veldkamp F. D.: Octonions, Jordan Algebras and Exceptional Groups.Springer, Berlin, 2000. Zbl 1087.17001, MR 1763974 |
| Reference:
|
[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). MR 1699860 |
| . |
