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.

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.

1 Algebra
2 References

[edit] Algebra

[edit] Dynkin diagram

[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

John Baez, The Octonions, Section 4.2: F₄, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at

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

Exceptional Lie groups

E₆ | E₇ | E₈ | F₄ | G₂

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Category: Lie groups

F4 (mathematics)

From Wikipedia, the free encyclopedia

Contents

[edit] Algebra

[edit] Dynkin diagram

[edit] Roots of F₄

[edit] Weyl/Coxeter group

[edit] Cartan matrix

[edit] F₄ lattice

[edit] References

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Navigation

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In other languages

F4 (mathematics)

From Wikipedia, the free encyclopedia

Contents

[edit] Algebra

[edit] Dynkin diagram

[edit] Roots of F4

[edit] Weyl/Coxeter group

[edit] Cartan matrix

[edit] F4 lattice

[edit] References

Views

Navigation

interaction

Search

In other languages

[edit] Roots of F₄

[edit] F₄ lattice