The geometry of null systems, Jordan algebras and von Staudt's theorem
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, p. 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 : https://doi.org/10.5802/aif.1942
Classification:  17C37,  51A05,  51A50,  51N25,  53C35
Keywords: null-system, projective geometry, polar geometry, symmetric space, Jordan algebra
@article{AIF_2003__53_1_193_0,
     author = {Bertram, Wolfgang},
     title = {The geometry of null systems, Jordan algebras and von Staudt's theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {1},
     year = {2003},
     pages = {193-225},
     doi = {10.5802/aif.1942},
     zbl = {1038.17023},
     mrnumber = {1973071},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_1_193_0}
}
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. http://www.numdam.org/item/AIF_2003__53_1_193_0/

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