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

Bertram, Wolfgang

Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 193-225.

Nous caractérisons une classe importante de géométries projectives généralisées $(X,X^{\text{'}})$ par les propriétés équivalentes suivantes : (1) $(X,X^{\text{'}})$ admet une polarité nulle centrale; (2) $(X,X^{\text{'}})$ admet une polarité intérieure; (3) $(X,X^{\text{'}})$ 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.

We characterize an important class of generalized projective geometries $(X,X^{\text{'}})$ by the following essentially equivalent properties: (1) $(X,X^{\text{'}})$ admits a central null-system; (2) $(X,X^{\text{'}})$ admits inner polarities: (3) $(X,X^{\text{'}})$ 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.

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

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