F4 (mathematics)

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The correct title of this article is F4 (mathematics). It features superscript or subscript characters that are substituted or omitted because of technical limitations.
Graph of 24-cell regular polytopeCoxeter-Dynkin diagram:
Graph of 24-cell regular polytope

Coxeter-Dynkin diagram: Image:CDW_ring.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.png

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. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the 'octonionic projective plane', OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

There are 3 real forms: a compact one, a split one, and a third one.

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.


Groups
Group theory
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Contents

[edit] Algebra

[edit] Dynkin diagram

Dynkin diagram of F_4

[edit] 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)
\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)

Simple roots

(0,1, − 1,0)
(0,0,1, − 1)
(0,0,0,1)
\left(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)

[edit] Weyl/Coxeter group

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

[edit] Cartan matrix

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

[edit] 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. The 24 Hurwitz quaternions of norm 1 form the 24-cell.

[edit] References

http://math.ucr.edu/home/baez/octonions/node15.html.

Exceptional Lie groups
E6 | E7 | E8 | F4 | G2
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