nLab
F4

Context

Exceptional structures

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

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 4F_4:

Aut(Mat 3×3 herm(𝕆),)F 4. Aut\left( Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4 \,.

(e.g. Yokota 09, section 2.2)

References

General

Cohomological properties:

In string theory

That the group F 4F_4 controls the massless degrees of freedom of 11-dimensional supergravity was observed and explored in