F4 (mathematics) - Definition and Overview

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.

Algebra

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.

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

Simple roots

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

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

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

<math>(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})<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.