We characterize an important class of generalized projective geometries by the following essentially equivalent properties: (1) admits a central null-system; (2) admits inner polarities: (3) 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 par les propriétés équivalentes suivantes : (1) admet une polarité nulle centrale; (2) admet une polarité intérieure; (3) 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.
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{\textquoteright}s} theorem}, journal = {Annales de l'Institut Fourier}, pages = {193--225}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {1}, year = {2003}, doi = {10.5802/aif.1942}, mrnumber = {1973071}, zbl = {1038.17023}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1942/} }
TY - JOUR TI - The geometry of null systems, Jordan algebras and von Staudt’s theorem JO - Annales de l'Institut Fourier PY - 2003 DA - 2003/// SP - 193 EP - 225 VL - 53 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1942/ UR - https://www.ams.org/mathscinet-getitem?mr=1973071 UR - https://zbmath.org/?q=an%3A1038.17023 UR - https://doi.org/10.5802/aif.1942 DO - 10.5802/aif.1942 LA - en ID - AIF_2003__53_1_193_0 ER -
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/
[Ar66] Geometric Algebra, Interscience, New York, 1966 | MR: 82463 | Zbl: 0077.02101
[B94] Geometry Tome 2 volumes, Springer-Verlag, Berlin, 1994 | MR: 1295239 | Zbl: 0606.51001
[Be00] The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, Tome 1754, Springer, Berlin, 2000 | MR: 1809879 | Zbl: 1014.17024
[Be01a] From linear algebra via affine algebra to projective algebra (2001) (preprint, Nancy) | MR: 2031788 | Zbl: 1063.17020
[Be01b] 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] Jordan-Algebren, Springer-Verlag, Berlin, 1965 | MR: 204470 | Zbl: 0145.26001
[Br68] Doppelverhältnisse in Jordan-Algebren, Abh. Math. Sem. Hamburg, Tome 32 (1968), pp. 25-51 | Article | MR: 233858 | Zbl: 0175.31101
[Ch49] On the geometry of algebraic homogeneous spaces, Ann. Math, Tome 50 (1949) no. 1, pp. 32-67 | Article | MR: 28057 | Zbl: 0040.22901
[FK94] Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994 | MR: 1446489 | Zbl: 0841.43002
[Hua45] 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] 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] Gruppen und Lie-Algebren von rationalen Funktionen, Math. Z, Tome 109 (1969), pp. 349-392 | Article | MR: 251092 | Zbl: 0181.04503
[Lo69] Symmetric Spaces I, Benjamin, New York, 1969 | Zbl: 0175.48601
[Lo75] Jordan Pairs, LN, Tome 460, Springer, New York, 1975 | MR: 444721 | Zbl: 0301.17003
[Lo95] Elementary Groups and Stability for Jordan Pairs, K-Theory, Tome 9 (1995), pp. 77-116 | Article | MR: 1340841 | Zbl: 0835.17021
[Sp73] Jordan Algebras and Algebraic Groups, Springer Verlag, New York, 1973 | MR: 379618 | Zbl: 0259.17003
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