Inner ideals and intrinsic subspaces
We introduce the notion of intrinsic subspaces of
linear and affine pair geometries, which generalizes
the one of projective subspaces of projective spaces.
We prove that, when the affine pair geometry is the projective
geometry of a Lie algebra introduced in [Bertram-Neeb, J. Alg. 277],
such intrinsic
subspaces correspond to inner ideals in the associated Jordan
pair, and we investigate the case of intrinsic subspaces defined
by the Peirce-decomposition which is related to $5$-gradings of the
projective Lie algebra. These examples, as well as the examples of
general and Lagrangian flag geometries, lead to the
conjecture that geometries of
intrinsic subspaces tend to be themselves linear pair geometries.
(To appear in Adv. Geom.)
W. Bertram < bertram@iecn.u-nancy.fr >
H. Loewe < h.loewe@tu-bs.de >