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>