nLab F₄

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)

Related concepts

• exceptional geometry, exceptional generalized geometry

• G₂, F₄,

  E₆, E₇, E₈, E₉, E₁₀, E₁₁, $\cdots$

References

General

• José Figueroa-O'Farrill, A geometric construction of the exceptional Lie algebras $F_4$ and $E_8$ (arXiv:0706.2829)

• Ichiro Yokota, Exceptional Lie groups (arXiv:0902.0431)

Cohomological properties:

• Y. Choi, 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)

See also:

• V.K. Dobrev: Invariant Differential Operators for the Real Exceptional Lie Algebra $F'_4$ [arXiv:2503.22407]

In string theory

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

• T. Pengpan, Pierre Ramond, M(ysterious) Patterns in SO(9), Phys. Rept. 315:137-152,1999 (arXiv:hep-th/9808190)

• Pierre Ramond, Boson-Fermion Confusion: The String Path To Supersymmetry, Nucl.Phys.Proc.Suppl. 101 (2001) 45-53 (arXiv:hep-th/0102012)

• Pierre Ramond, Algebraic Dreams (arXiv:hep-th/0112261)

• Hisham Sati, $\mathbb{O}P^2$-bundles in M-theory, Commun. Num. Theor. Phys 3:495-530, 2009 (arXiv:0807.4899)