The geometry of null systems, Jordan algebras and von Staudt's theorem

Wolfgang Bertram[1]

  • [1] Université Nancy I, Institut Élie Cartan, BP 239, 54506 Vandoeuvre-les-Nancy Cedex (France)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 1, page 193-225
  • ISSN: 0373-0956

Abstract

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We characterize an important class of generalized projective geometries ( X , X ' ) by the following essentially equivalent properties: (1) ( X , X ' ) admits a central null-system; (2) ( X , X ' ) admits inner polarities: (3) ( X , X ' ) is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s theorem which generalizes similar results by L.K. Hua.

How to cite

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Bertram, Wolfgang. "The geometry of null systems, Jordan algebras and von Staudt's theorem." Annales de l’institut Fourier 53.1 (2003): 193-225. <http://eudml.org/doc/116033>.

@article{Bertram2003,
abstract = {We characterize an important class of generalized projective geometries $(X,X^\{\prime \})$ by the following essentially equivalent properties: (1) $(X,X^\{\prime \})$ admits a central null-system; (2) $(X,X^\{\prime \})$ admits inner polarities: (3) $(X,X^\{\prime \})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s theorem which generalizes similar results by L.K. Hua.},
affiliation = {Université Nancy I, Institut Élie Cartan, BP 239, 54506 Vandoeuvre-les-Nancy Cedex (France)},
author = {Bertram, Wolfgang},
journal = {Annales de l’institut Fourier},
keywords = {null-system; projective geometry; polar geometry; symmetric space; Jordan algebra},
language = {eng},
number = {1},
pages = {193-225},
publisher = {Association des Annales de l'Institut Fourier},
title = {The geometry of null systems, Jordan algebras and von Staudt's theorem},
url = {http://eudml.org/doc/116033},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Bertram, Wolfgang
TI - The geometry of null systems, Jordan algebras and von Staudt's theorem
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 1
SP - 193
EP - 225
AB - We characterize an important class of generalized projective geometries $(X,X^{\prime })$ by the following essentially equivalent properties: (1) $(X,X^{\prime })$ admits a central null-system; (2) $(X,X^{\prime })$ admits inner polarities: (3) $(X,X^{\prime })$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s theorem which generalizes similar results by L.K. Hua.
LA - eng
KW - null-system; projective geometry; polar geometry; symmetric space; Jordan algebra
UR - http://eudml.org/doc/116033
ER -

References

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  11. P. Jordan J. von Neumann, E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math 35 (1934), 29-64 Zbl0008.42103MR1503141
  12. M. Koecher, Gruppen und Lie-Algebren von rationalen Funktionen, Math. Z 109 (1969), 349-392 Zbl0181.04503MR251092
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  16. T.A. Springer, Jordan Algebras and Algebraic Groups, (1973), Springer Verlag, New York Zbl0259.17003MR379618

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