From linear algebra via affine algebra to projective algebra
In his fundamental paper [``Generalized projective geometries: General theory and equivalence with Jordan structures'', Adv. Geom. 2, No. 4, 329--369 (2002; Zbl 1035.17043)], the author introduced \textit{generalized projective spaces} as the global geometric objects belonging to Jordan pairs over arbitrary base rings. One basic step on the way to the axiomatic description of these objects is contained in the paper under review: Every affine space over a field of characteristic \(\not=2\) can be described in terms of the ``product maps'' \((x,y) \mapsto (1-r)x+ry\). The main part of this paper is devoted to important classical examples of generalized projective geometries: projective geometries, Grassmannian geometries, Lagrangian geometries and conformal geometries (of arbitrary dimensions and over arbitrary communtative base rings, provided that \(2\) is invertible) are discussed here. Moreover, the author considers polarities of these geometries and gives links to the theory of symmetric spaces.
scientific article; zbMATH DE number 2081750
scientific article