This entry is about items in the ADE-classification labeled by $D4$. For the D4-brane, see there.
In the ADE-classification, the items labeled $D_4$ include the following:
as Dynkin diagram/Dynkin quiver:
(graphics grabbed from Wikipedia here)
the Klein four-group (the smallest dihedral group)
$\mathbb{Z}/2 \times \mathbb{Z}/2$
the quaternion group of order 8 (the smallest binary dihedral group):
$Q_8 \simeq 2 D_4$
as simple Lie groups: the special orthogonal group in 8 dimensions
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 solid | finite 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 $SU(n+1)$ | |
D4 | Klein 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 $SO(2n)$ |
$E_6$ | tetrahedron | tetrahedral 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.