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

[La géométrie des polarités nulles, algèbres de Jordan et le théorème de von Staudt]

Bertram, Wolfgang

Annales de l'Institut Fourier, Tome 53 (2003), p. 193-225 / Harvested from Numdam

Résumé

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

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.

Détail

Publié le : 2003-01-01

MR 1973071 | Zbl 1038.17023

DOI : https://doi.org/10.5802/aif.1942
Classification:  17C37,  51A05,  51A50,  51N25,  53C35
Mots clés: polarité nulle, géométrie projective, géométrie polaire, espace symétriques, algèbre de Jordan

Bibliographie

[Ar66] E. Artin Geometric Algebra, Interscience, New York (1966) | MR 82463 | Zbl 0077.02101

[B94] M. Berger Geometry, Springer-Verlag, Berlin Tome 2 volumes (1994) | MR 1295239 | Zbl 0606.51001

[Be00] W. Bertram The geometry of Jordan and Lie structures, Springer, Berlin, Lecture Notes in Mathematics, Tome 1754 (2000) | MR 1809879 | Zbl 1014.17024

[Be01a] W. Bertram From linear algebra via affine algebra to projective algebra (2001) (preprint, Nancy) | MR 2031788 | Zbl 1063.17020

[Be01b] W. Bertram Generalized projective geometries: general theory and equivalence with Jordan structures (2001) (preprint, Nancy (to appear in Advances in Geometry)) | MR 2070568 | Zbl 1035.17043

[BK65] H. Braun; M. Koecher Jordan-Algebren, Springer-Verlag, Berlin (1965) | MR 204470 | Zbl 0145.26001

[Br68] H. Braun Doppelverhältnisse in Jordan-Algebren, Abh. Math. Sem. Hamburg, Tome 32 (1968), pp. 25-51 | Article | MR 233858 | Zbl 0175.31101

[Ch49] W.L. Chow On the geometry of algebraic homogeneous spaces, Ann. Math, Tome 50 (1949) no. 1, pp. 32-67 | Article | MR 28057 | Zbl 0040.22901

[FK94] J. Faraut.; A. Koranyi Analysis on Symmetric Cones, Clarendon Press, Oxford (1994) | MR 1446489 | Zbl 0841.43002

[Hua45] L.-K. Hua Geometries of Matrices. I. Generalizations of von Staudt's theorem, Trans. A.M.S, Tome 57 (1945), pp. 441-481 | MR 12679 | Zbl 0063.02922

[JNW34] P. Jordan J. Von Neumann; E. Wigner On an algebraic generalization of the quantum mechanical formalism, Ann. Math, Tome 35 (1934), pp. 29-64 | Article | JFM 60.0902.02 | MR 1503141 | Zbl 0008.42103

[Koe69] M. Koecher Gruppen und Lie-Algebren von rationalen Funktionen, Math. Z, Tome 109 (1969), pp. 349-392 | Article | MR 251092 | Zbl 0181.04503

[Lo69] O. Loos Symmetric Spaces I, Benjamin, New York (1969) | Zbl 0175.48601

[Lo75] O. Loos Jordan Pairs, Springer, New York, LN, Tome 460 (1975) | MR 444721 | Zbl 0301.17003

[Lo95] O. Loos Elementary Groups and Stability for Jordan Pairs, K-Theory, Tome 9 (1995), pp. 77-116 | Article | MR 1340841 | Zbl 0835.17021

[Sp73] T.A. Springer Jordan Algebras and Algebraic Groups, Springer Verlag, New York (1973) | MR 379618 | Zbl 0259.17003