Recovering the Lie algebra from its extremal geometry

TL;DR

This paper investigates the extremal geometry of simple Lie algebras, showing that in simply-laced types, the algebra can be reconstructed as a quotient of a Chevalley algebra based on geometric properties.

Contribution

It establishes a connection between extremal geometries and the structure of Lie algebras, proving that simply-laced types are quotients of Chevalley algebras.

Findings

Extremal geometry is a subspace of the projective geometry of the Lie algebra.

For simply-laced types, the Lie algebra is a quotient of a Chevalley algebra.

The extremal geometry corresponds to the root shadow space of an irreducible spherical building.

Abstract

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $\Delta$. We prove that if $\Delta$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type.