Generalized projective geometries: general theory and equivalence with Jordan structures

In this work we introduce generalized projective geometries which are a natural generalization of projective geometries over a field or ring K but also of other important geometries such as Grassmannian, Lagrangian or conformal geometry (see [3]). We also introduce the corresponding generalized polar geometries and associate to such a geometry a symmetric space over K. In the finite-dimensional case over K 1⁄4 R, 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, respectively to 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.