Generalized projective geometries. Part II: General theory and equivalence with Jordan structures.

In this work we introduce generalized projective geometries which are a natural generalization of projective geometry over a field or ring K but also of other important geometries such as Grassmannian, Lagrangian or conformal geometry (see Part I). We also introduce the corresponding generalized polar geometries and associate to such a geometry a symmetric space over K. In the finite-dimensional over the real numbers, all classical and many exceptional symmetric spaces are obtained in this way. We prove that generalized projective and polar geometries are essentially equivalent to Jordan algebraic structures, namely to Jordan pairs, resp. Jordan triple systems over K which are obtained as a linearized tangent version of the geometries in a similar way as a Lie group is linearized by its Lie algebra. In contrast to the case of Lie theory, the construction of the "Jordan functor" works equally well over general base rings and in arbitrary dimension.

(This paper has appearde in Adv. Geom. 2 (2002), 329--369)


W. Bertram <bertram@iecn.u-nancy.fr>