The geometry of null systems, Jordan algebras and von Staudt's theorem
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 193-225.

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.

Nous caractérisons une classe importante de géométries projectives généralisées (X,X ' ) par les propriétés équivalentes suivantes : (1) (X,X ' ) admet une polarité nulle centrale; (2) (X,X ' ) admet une polarité intérieure; (3) (X,X ' ) est associée à une algèbre de Jordan avec élément neutre. Dans ce cadre, nous démontrons un analogue du théorème de von Staudt qui généralise des résultats similaires de L.K. Hua.

DOI: 10.5802/aif.1942
Classification: 17C37, 51A05, 51A50, 51N25, 53C35
Keywords: null-system, projective geometry, polar geometry, symmetric space, Jordan algebra
Mot clés : polarité nulle, géométrie projective, géométrie polaire, espace symétriques, algèbre de Jordan
Bertram, Wolfgang 1

1 Université Nancy I, Institut Élie Cartan, BP 239, 54506 Vandoeuvre-les-Nancy Cedex (France)
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Bertram, Wolfgang. The geometry of null systems, Jordan algebras and von Staudt's theorem. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 193-225. doi : 10.5802/aif.1942. https://aif.centre-mersenne.org/articles/10.5802/aif.1942/

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