F4 (mathematics)

Encyclopedia : F : F4 : F4M : F4 (mathematics)

In mathematics, F₄ is the name of a Lie group and also its Lie algebra [\mathfrak_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.

Contents

1 Algebra

1.1
1.2 Roots of F₄
1.3 Weyl/Coxeter group
1.4
1.5 F₄ lattice

2 References

Algebra

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,\pm\frac,\pm\frac,\pm\frac\right)]

Simple roots

[(0,0,0,1)]

[(0,0,1,-1)]

[(0,1,-1,0)]

[\left(\frac,-\frac,-\frac,-\frac\right)]

Weyl/Coxeter group

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

[\begin2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end]

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.

Exceptional Lie groups
E₆ | E₇ | E_7½ | E₈ | F₄ | G₂

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.