On exceptional Lie geometries

TL;DR

This paper characterizes a broad class of Lie geometries, including many exceptional types, as parapolar spaces with a specific intersection property, enhancing understanding of their geometric structure.

Contribution

It provides a new characterization of Lie geometries via parapolar spaces with a simple intersection property, extending to locally disconnected cases.

Findings

Many exceptional Lie incidence geometries are characterized as parapolar spaces.

The characterization applies to both connected and locally disconnected geometries.

The results unify various known characterizations of Lie geometries.

Abstract

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular many of the exceptional Lie incidence geometries occur. {In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well known characterizations.