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Zbl 1038.17023
Bertram, Wolfgang
The geometry of null systems, Jordan algebras and von Staudt's theorem. (English)
[J] Ann. Inst. Fourier 53, No. 1, 193-225 (2003). ISSN 0373-0956; ISSN 1777-5310/e

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 [{\it M. Berger}, Geometry I, II. Berlin: Springer-Verlag (1987; Zbl 0606.51001), Reprint Springer (1994)].
[Demetra Demetropoulou-Psomopoulou (Thessaloniki)]

MSC 2000:: *17C37 Associated geometries
51A05 General theory of linear incidence geometry
53C35 Symmetric spaces (differential geometry)

Keywords: null-system; projective geometry; polar geometry; symmetric space; Jordan algebra

Citations: Zbl 1035.17043; Zbl 0606.51001

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