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.
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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
Dynkin_diagram_F4.png
Dynkin diagram of F_4
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.
| E6 | E7 | E8 | F4 | G2 |