Inner ideals in Lie algebras and spherical buildings

TL;DR

This paper extends Faulkner's correspondence between inner ideals of Lie algebras and shadows on root groups to a broader class of algebraic groups over perfect fields with characteristic not two.

Contribution

It generalizes the known correspondence to more general algebraic groups and fields, broadening the theoretical framework connecting Lie algebra ideals and geometric structures.

Findings

The correspondence holds over perfect fields of characteristic not two.

The relationship between inner ideals and building shadows is established in greater generality.

The results unify algebraic and geometric perspectives in Lie theory.

Abstract

The correspondence found by Faulkner between inner ideals of the Lie algebra of a simple algebraic group and shadows on long root groups of the building associated with the algebraic group is shown to hold in greater generality (in particular, over perfect fields of characteristic distinct from two).