F4 (mathematics)

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. F₄ 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 F₄ 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

Dynkin diagram of F_4

Roots of F₄

$(\pm 1,\pm 1,0,0)$

$(\pm 1,0,\pm 1,0)$

$(\pm 1,0,0,\pm 1)$

$(0,\pm 1,\pm 1,0)$

$(0,\pm 1,0,\pm 1)$

$(0,0,\pm 1,\pm 1)$

$(\pm 1,0,0,0)$

$(0,\pm 1,0,0)$

$(0,0,\pm 1,0)$

$(0,0,0,\pm 1)$

$\left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\pm {\frac {1}{2}},\pm {\frac {1}{2}}\right)$

Simple roots

$(0,0,0,1)$

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

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

$\left({\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {1}{2}}\right)$

Weyl/Coxeter group

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

Cartan matrix

${\begin{pmatrix}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{pmatrix}}$

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.

References

• John Baez, The Octonions, Section 4.2: F₄, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at

http://math.ucr.edu/home/baez/octonions/node15.html.