The exceptional simple Lie group $F_{4(-20)}$, after J. Tits

TL;DR

This paper explores the structure and automorphisms of the exceptional Lie group F4(-20), focusing on Tits' construction of the hyperbolic plane over Cayley numbers and explicit group actions.

Contribution

It explicitly describes the action of the unipotent subgroup N on the hyperbolic plane and characterizes the minimal parabolic subgroup M geometrically.

Findings

Explicit description of N's action on H^2(Cay)

Geometric characterization of M

Insights into Tits' synthetic construction

Abstract

This is a semi-survey paper, where we start by advertising Tits' synthetic construction from \cite{Tits}, of the hyperbolic plane $H^2(Cay)$ over the Cayley numbers $Cay$, and of its automorphism group which is the exceptional simple Lie group $G=F_{4(-20)}$. Let $G=KAN$ be the Iwasawa decomposition. Our contributions are: a) Writing down explicitly the action of $N$ on $H^2(Cay)$ in Tits'model, facing the lack of associativity of $Cay$. b) If $MAN$ denotes the minimal parabolic subgroup of $G$, characterizing $M$ geometrically.