F₄

In mathematics, F₄ is the name of a Lie group and also its Lie algebra $$mathfrak{f}_4. It is one of the five exceptional simple Lie groups. "F"₄ has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

The compact real form of F₄ is the isometry group of a 16-dimensional Riemannian manifold known as the 'octonionic projective plane', OP². This can be seen systematically using a construction known as the "magic square", due to Hans Freudenthal and Jacques Tits.

There are 3 real forms: a compact one, a split one, and a third one.

The "F"₄ Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E₈.

Algebra

Dynkin diagram

Roots of F₄

:$$(pm 1,pm 1,0,0)

:$$(pm 1,0,pm 1,0)

:$$(pm 1,0,0,pm 1)

:$$(0,pm 1,pm 1,0)

:$$(0,pm 1,0,pm 1)

:$$(0,0,pm 1,pm 1)

:$$(pm 1,0,0,0)

:$$(0,pm 1,0,0)

:$$(0,0,pm 1,0)

:$$(0,0,0,pm 1)

:$$left(pmfrac{1}{2},pmfrac{1}{2},pmfrac{1}{2},pmfrac{1}{2} ight)

Simple roots:$$(0,1,-1,0)

:$$(0,0,1,-1)

:$$(0,0,0,1)

:$$left(frac{1}{2},-frac{1}{2},-frac{1}{2},-frac{1}{2} ight)

Weyl/Coxeter group

Its Weyl/Coxeter group is the symmetry group of the 24-cell.

Cartan matrix

:$$egin{pmatrix}2&-1&0&0\-1&2&-2&0\0&-1&2&-1\0&0&-1&2end{pmatrix}

F₄ lattice

The F₄ lattice is a four dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the 24-cell.

ee also

* Exceptional Jordan algebra

References

* John Baez, "The Octonions", Section 4.2: F₄, [http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html Bull. Amer. Math. Soc. 39 (2002), 145-205] . Online HTML version at http://math.ucr.edu/home/baez/octonions/node15.html.