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F4 (mathematics)

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

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.

Roots of F₄

$(\pm 1,\pm 1,0,0)$

$(\pm 1,0,\pm 1,0)$

$(\pm 1,0,0,\pm 1)$

$(0,\pm 1,\pm 1,0)$

$(0,\pm 1,0,\pm 1)$

$(0,0,\pm 1,\pm 1)$

$(\pm 1,0,0,0)$

$(0,\pm 1,0,0)$

$(0,0,\pm 1,0)$

$(0,0,0,\pm 1)$

$(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})$

Simple roots

(0,0,0,1)

(0,0,1, - 1)

(0,1, - 1,0)

$(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$

Weyl/Coxeter group

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

Cartan matrix

$\begin{pmatrix} 2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2 \end{pmatrix}$

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.

Categories: Lie groups

Last updated: 05-24-2005 20:21:36

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10-26-2009 08:16:03
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