# nLab F4

Contents

## Philosophy

group theory

### Cohomology and Extensions

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

One of the exceptional Lie groups.

## Definition

###### Definition/Proposition

(Jordan algebra automorphism group of octonionic Albert algebra is F4)

The group of automorphism with respect to the Jordan algebra structure $\circ$ on the octonionic Albert algebra is the exceptional Lie group $F_4$:

$Aut\left( Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4 \,.$

(e.g. Yokota 09, section 2.2)

• G2, F4,

E6, E7, E8, E9, E10, E11, $\cdots$

## References

### General

Cohomological properties:

• Y. Cchoi, S. Yoon, Homology of the triple loop space of the exceptional Lie group $F_4$, J. Korean Math. Soc. 35 (1998), No. 1, pp. 149–164 (pdf)

### In string theory

That the group $F_4$ controls the massless degrees of freedom of 11-dimensional supergravity was observed and explored in

Last revised on May 14, 2019 at 04:51:19. See the history of this page for a list of all contributions to it.