×

The geometry of null systems, Jordan algebras and von Staudt’s theorem. (English) Zbl 1038.17023

The paper under review extends definitions and results on generalized projective and polar geometry and their equivalence with the Jordan structures of the author’s recent paper [Adv. Geom. 2, 329–369 (2002; Zbl 1035.17043)]. The main topic of this work is that the generalization of a connected generalized projective geometry \((X, X')\) over a commutative ring \(K\) with unit \(1\) and \({1\over 2}\in K\), is given by spaces corresponding to unital Jordan algebras, that is Jordan pair together with a distinguished invertible element. Also, the geometric interpretation of unital Jordan algebras is discussed. Precisely, there is canonically associated to the geometry \((X, X')\) a class of symmetric spaces. Additionally, a generalization is given to the well-known von Staudt’s theorem [M. Berger, Geometry I, II. Berlin: Springer-Verlag (1987; Zbl 0606.51001), Reprint Springer (1994)].

MSC:

17C37 Associated geometries of Jordan algebras
51A05 General theory of linear incidence geometry and projective geometries
53C35 Differential geometry of symmetric spaces
PDFBibTeX XMLCite
Full Text: DOI Numdam Numdam EuDML

References:

[1] Geometric Algebra (1966) · Zbl 0077.02101
[2] Geometry, 2 volumes (1994) · Zbl 0606.51001
[3] The geometry of Jordan and Lie structures, 1754 (2000) · Zbl 1014.17024
[4] From linear algebra via affine algebra to projective algebra (2001) · Zbl 1063.17020
[5] Generalized projective geometries: general theory and equivalence with Jordan structures (2001) · Zbl 1035.17043
[6] Jordan-Algebren (1965) · Zbl 0145.26001
[7] Doppelverhältnisse in Jordan-Algebren, Abh. Math. Sem. Hamburg, 32, 25-51 (1968) · Zbl 0175.31101
[8] On the geometry of algebraic homogeneous spaces, Ann. Math, 50, 1, 32-67 (1949) · Zbl 0040.22901
[9] Analysis on Symmetric Cones (1994) · Zbl 0841.43002
[10] Geometries of Matrices. I. Generalizations of von Staudt’s theorem, Trans. A.M.S, 57, 441-481 (1945) · Zbl 0063.02922
[11] On an algebraic generalization of the quantum mechanical formalism, Ann. Math, 35, 29-64 (1934) · JFM 60.0902.02
[12] Gruppen und Lie-Algebren von rationalen Funktionen, Math. Z, 109, 349-392 (1969) · Zbl 0181.04503
[13] Symmetric Spaces I (1969) · Zbl 0175.48601
[14] Jordan Pairs, 460 (1975) · Zbl 0301.17003
[15] Elementary Groups and Stability for Jordan Pairs, K-Theory, 9, 77-116 (1995) · Zbl 0835.17021
[16] Jordan Algebras and Algebraic Groups (1973) · Zbl 0259.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.