A geometric characterization of the classical Lie algebras

TL;DR

This paper demonstrates that the structure of classical Lie algebras can be uniquely determined by their extremal geometry, linking algebraic properties to geometric configurations, especially for buildings of rank at least 3.

Contribution

It establishes that the extremal geometry uniquely characterizes classical Lie algebras with high-rank buildings, extending previous geometric characterizations.

Findings

Extremal geometry is a root shadow space of a spherical building.

The isomorphism type of a Lie algebra is determined by its extremal geometry for rank ≥ 3.

Provides a geometric characterization of classical Lie algebras.

Abstract

A nonzero element x in a Lie algebra g over a field F with Lie product [ , ] is called a extremal element if [x, [x, g]] is contained in Fx. Long root elements in classical Lie algebras are examples of extremal elements. Arjeh Cohen et al. initiated the investigation of Lie algebras generated by extremal elements in order to provide a geometric characterization of the classical Lie algebras generated by their long root el- ements. He and Gabor Ivanyos studied the so-called extremal geometry with as points the 1-dimensional subspaces of g generated by extremal elements of g and as lines the 2-dimensional subspaces of g all whose nonzero vectors are extremal. For simple finite dimensional g this geometry turns out to be a root shadow space of a spherical building. In this paper we show that the isomorphism type of g is determined by its extremal geometry, provided the building has rank at least 3.