D4 (changes) in nLab

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

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Contents

Idea

In the ADE-classification, the items labeled D 4D4 D_4 D4 include the following:

1. as finite subgroups of SO(3):

   the Klein four-group (the smallest dihedral group)

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

2. as finite subgroups of SU(2):

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

   Q 8≃2D 4Q_8 \simeq 2 D_4

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

   SO(8), Spin(8)

4. as a Dynkin diagram/Dynkin quiver:

\begin{tikzpicture} \node (center) at (0,0) {}; \node (topright) at (60:1) {}; \node (botright) at (-60:1) {}; \node (left) at (180:1) {};

\drawfill=black circle (.1); \drawfill=black circle (.1); \drawfill=black circle (.1); \drawfill=black circle (.1);

\draw (center) to (topright); \draw (center) to (botright); \draw (center) to (left); \end{tikzpicture}

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	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
A n≥1A_{n \geq 1}		cyclic group ℤ n+1\mathbb{Z}_{n+1}	cyclic group ℤ n+1\mathbb{Z}_{n+1}	special unitary group SU(n+1)SU(n+1)
A1		cyclic group of order 2 ℤ 2\mathbb{Z}_2	cyclic group of order 2 ℤ 2\mathbb{Z}_2	SU(2)
A2		cyclic group of order 3 ℤ 3\mathbb{Z}_3	cyclic group of order 3 ℤ 3\mathbb{Z}_3	SU(3)
A3 = D3		cyclic group of order 4 ℤ 4\mathbb{Z}_4	cyclic group of order 4 2D 2≃ℤ 42 D_2 \simeq \mathbb{Z}_4	SU(4) ≃\simeq Spin(6)
D4	dihedron on bigon	Klein four-group D 4≃ℤ 2×ℤ 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2	quaternion group 2D 4≃2 D_4 \simeq Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 D 6D_6	binary dihedral group of order 12 2D 62 D_6	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 D 8D_8	binary dihedral group of order 16 2D 82 D_{8}	SO(12), Spin(12)
D n≥4D_{n \geq 4}	dihedron, hosohedron	dihedral group D 2(n−2)D_{2(n-2)}	binary dihedral group 2D 2(n−2)2 D_{2(n-2)}	special orthogonal group, spin group SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6	tetrahedron	tetrahedral group TT	binary tetrahedral group 2T2T	E6
E 7E_7	cube, octahedron	octahedral group OO	binary octahedral group 2O2O	E7
E 8E_8	dodecahedron, icosahedron	icosahedral group II	binary icosahedral group 2I2I	E8