On the geometry of inner ideals

Abstract

The notion of an inner ideal, which has arisen in the study of Jordan algebras, is extended here to an arbitrary finite dimensional module M for a finite dimensional Lie algebra with nondegenerate symmetric associative bilinear form. The extension is made by first defining a product xy ∗z for x, z ϵ M , y ∗ ϵ M ∗ (the contragredient module). With suitable identification of M ∗ with M , this product is, in various cases, that of a Jordan triple system, a Lie triple system, a symplectic ternary algebra, and other ternary algebras. The inner ideals of M are used to describe several special geometries previously defined by ad hoc methods on certain Lie modules. Finally, for a split semisimple Lie algebra over a field of characteristic zero, the inner ideals are shown to correspond to the objects of a geometry defined by Tits from the corresponding Chevalley group.