# F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra [itex]\mathfrak{f}_4[itex]. 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.

 Contents

## 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

Missing image
Dynkin_diagram_F4.png
Dynkin diagram of F_4

### Roots of F4

[itex](\pm 1,\pm 1,0,0)[itex]
[itex](\pm 1,0,\pm 1,0)[itex]
[itex](\pm 1,0,0,\pm 1)[itex]
[itex](0,\pm 1,\pm 1,0)[itex]
[itex](0,\pm 1,0,\pm 1)[itex]
[itex](0,0,\pm 1,\pm 1)[itex]
[itex](\pm 1,0,0,0)[itex]
[itex](0,\pm 1,0,0)[itex]
[itex](0,0,\pm 1,0)[itex]
[itex](0,0,0,\pm 1)[itex]
[itex](\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})[itex]

Simple roots

[itex](0,0,0,1)[itex]
[itex](0,0,1,-1)[itex]
[itex](0,1,-1,0)[itex]
[itex](\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})[itex]

### Weyl/Coxeter group

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

### Cartan matrix

[itex]

\begin{pmatrix} 2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2 \end{pmatrix} [itex]

### 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.

 E6 | E7 | E8 | F4 | G2

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