nLab F₄

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Contents

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

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 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:

  • Y. Choi, S. Yoon, Homology of the triple loop space of the exceptional Lie group F 4F_4, J. Korean Math. Soc. 35 (1998), No. 1, pp. 149–164 (pdf)

In string theory

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

Last revised on August 8, 2024 at 20:08:40. See the history of this page for a list of all contributions to it.