# Mapping the geometry of the F4 group

## 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.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- Physics Division

- OSTI Identifier:
- 948502

- Report Number(s):
- LBNL-1517E

TRN: US0901666

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Advances in Theoretical and Mathematical Physics; Journal Volume: 12; Journal Issue: 4; Related Information: Journal Publication Date: 2008

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72; 97; ALGEBRA; CONSTRUCTION; GEOMETRY; JORDAN; LIE GROUPS; MATRICES; exceptional Lie groups, Haar measure, coset spaces, geometry

### Citation Formats

```
Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, and Cerchiai, Bianca L.
```*Mapping the geometry of the F4 group*. United States: N. p., 2007.
Web.

```
Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, & Cerchiai, Bianca L.
```*Mapping the geometry of the F4 group*. United States.

```
Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, and Cerchiai, Bianca L. Mon .
"Mapping the geometry of the F4 group". United States.
doi:. https://www.osti.gov/servlets/purl/948502.
```

```
@article{osti_948502,
```

title = {Mapping the geometry of the F4 group},

author = {Bernardoni, Fabio and Cacciatori, Sergio L and Scotti, Antonio and Cerchiai, Bianca L.},

abstractNote = {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.},

doi = {},

journal = {Advances in Theoretical and Mathematical Physics},

number = 4,

volume = 12,

place = {United States},

year = {Mon May 28 00:00:00 EDT 2007},

month = {Mon May 28 00:00:00 EDT 2007}

}