F4: Information from Answers.com

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.

Group theory

Group theory

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product

Discrete groups
Classification of finite simple groups Cyclic group Z_n Alternating group A_n Sporadic groups Mathieu group M_11..12,M_22..24 Conway group Co_1..3 Janko group J_{1, 2, 3, 4} Fischer group F_22..24 Baby Monster group B Monster group M Other finite groups Symmetric group, S_n Dihedral group, D_n Infinite groups The integers, Z Modular groups, PSL(2,Z) and SL(2,Z)

Continuous groups
Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G₂ F₄ E₆ E₇ E₈ Lorentz group Poincaré group

Infinite dimensional groups
conformal group diffeomorphism group Loop group Quantum group O(∞) SU(∞) Sp(∞)

1 Algebra
2 See also
3 References

Algebra

Dynkin diagram

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)

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

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

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.

v • d • e Exceptional Lie groups

E₆ \| E₇ \| E₈ \| F₄ \| G₂

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