nLab D4

Contents

This entry is about items in the ADE-classification labeled by $D4$. For the D4-brane, see there.

Contents

Idea

In the ADE-classification, the items labeled $D_4$ include the following:

1. (graphics grabbed from Wikipedia here)

2. the Klein four-group (the smallest dihedral group)

$\mathbb{Z}/2 \times \mathbb{Z}/2$

3. the quaternion group of order 8 (the smallest binary dihedral group):

$Q_8 \simeq 2 D_4$

4. as simple Lie groups: the special orthogonal group in 8 dimensions

SO(8)

Properties

Triality

The S3-symmetry group of the D4-diagram translates into interesting 3-fold symmetries of structures associated with the corresponding objects in the above list. This is known as triality.

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

Last revised on October 2, 2018 at 11:54:59. See the history of this page for a list of all contributions to it.