Mapping the geometry of the E6 group

TL;DR

This paper constructs a detailed parametrization of the exceptional Lie group E6 by exponentiating its Lie algebra, enabling explicit calculations such as the Haar measure, based on a fibration involving the F4 subgroup.

Contribution

It introduces a generalized Euler angle parametrization of E6 using a fibration over F4, extending previous work and providing explicit formulas for the Haar measure.

Findings

Explicit parametrization of E6 group manifold.

Derived the Haar invariant measure for E6.

Extended Euler angle approach to exceptional Lie groups.

Abstract

In this paper we present a construction for the compact form of the exceptional Lie group E6 by exponentiating the corresponding Lie algebra e6, which we realize as the the sum of f4, the derivations of the exceptional Jordan algebra J3 of dimension 3 with octonionic entries, and the right multiplication by the elements of J3 with vanishing trace. Our parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E6 via a F4 subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F4. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E6 group manifold.