nLab
D4

Contents

This entry is about items in the ADE-classification labeled by D4D4. For the D4-brane, see there.


Contents

Idea

In the ADE-classification, the items labeled D 4D_4 include the following:

  1. as Dynkin diagram/Dynkin quiver:

    (graphics grabbed from Wikipedia here)

  2. as finite subgroups of SO(3):

    the Klein four-group (the smallest dihedral group)

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

  3. as finite subgroups of SU(2):

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

    Q 82D 4Q_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.

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
D4Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group
SO(2n)SO(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
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.