Mapping the geometry of the F4 group

TL;DR

This paper constructs the compact form of the exceptional Lie group F4 by exponentiating its Lie algebra, providing explicit parametrization and measure, which aids in understanding F4's structure and its relation to E6.

Contribution

It presents a new explicit parametrization of F4 using a generalized Euler angle approach based on a Spin(9) fibration, and derives the Haar measure on F4.

Findings

Explicit parametrization of F4 group manifold.

Derived Haar invariant measure for F4.

Clarified the structure of F4 and its coset space OP2.

Abstract

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.