F4 (mathematics)

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In mathematics, F₄ is the name of a Lie group and also its Lie algebra <math>\mathfrak{f}_4</math>. It is one of the five exceptional simple Lie groups. F₄ has rank 4 and dimension 52. Its center is the trivial subgroup. 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.

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.

Contents 1 Algebra 1.1 Dynkin diagram 1.2 Roots of F4 1.3 Weyl/Coxeter group 1.4 Cartan matrix 1.5 F4 lattice 2 References

Algebra

Dynkin diagram

Roots of F₄

<math>(\pm 1,\pm 1,0,0)</math>

<math>(\pm 1,0,\pm 1,0)</math>

<math>(\pm 1,0,0,\pm 1)</math>

<math>(0,\pm 1,\pm 1,0)</math>

<math>(0,\pm 1,0,\pm 1)</math>

<math>(0,0,\pm 1,\pm 1)</math>

<math>(\pm 1,0,0,0)</math>

<math>(0,\pm 1,0,0)</math>

<math>(0,0,\pm 1,0)</math>

<math>(0,0,0,\pm 1)</math>

<math>\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)</math>

Simple roots

<math>(0,0,0,1)</math>

<math>(0,0,1,-1)</math>

<math>(0,1,-1,0)</math>

<math>\left(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)</math>

Weyl/Coxeter group

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

Cartan matrix

<math>

\begin{pmatrix} 2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2 \end{pmatrix} </math>

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.

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2