F4 (mathematics)

From Academic Kids

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.

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

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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.

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Dynkin diagram

Missing image
Dynkin_diagram_F4.png
Dynkin diagram of F_4

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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>
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Weyl/Coxeter group

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

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Cartan matrix

<math>

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

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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.

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2
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