Inner Ideals and Intrinsic Subspaces of Linear Pair Geometries

Wolfgang Bertram (1) , Harald Loewe (2)

1 IECN - Institut Élie Cartan de Nancy
2 Institut Comutational Mathematics

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.

Mots clés: intrinsic subspace; inner ideal; Jordan pair; graded and filtered Lie algebras; linear pair geometry

Domaines: Anneaux et algèbres [math.RA]

Wolfgang Bertram, Harald Loewe. Inner Ideals and Intrinsic Subspaces of Linear Pair Geometries. Advances in Geometry, 2007, 00 (00), pp.00. ⟨10.1515/ADVGEOM.2008.004⟩. ⟨hal-00080536⟩