nLab Platonic 2-group

Idea

A Platonic 2-group (Epa 10) is a 2-group higher extension of a “Platonic group” in the ADE classification, i.e. of a finite subgroup of SO(3); or else of its double cover, hence of finite subgroup of SU(2).

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
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	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

Properties

Relation to the string 2-group

Proposition

The universal Platonic 2-group extensions $\mathcal{G}_{uni}[i]$ of $G_{ADE}$ finite subgroups of SU(2) by $U(1)$ (Epa-Ganter 16, def.2.11) are equivalently the restrictions of the string 2-group extension $String(SU(2)) \to SU(2)$ of $SU(2)$, hence the classifying cocycle is the restriction of the smooth second Chern class:

$$\array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \big\downarrow && \big\downarrow && \big\downarrow^{\mathrlap{\mathbf{B}^3 i}} \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) } \,,$$

where $i$ is the canonical inclusion of roots of unity, $i: \mathbb{Z}/{|G|} \hookrightarrow U(1)$.

(Epa-Ganter 16, prop. 4.1)

See also at finite subgroup of SU(2) – Discrete torsion

References

• Narthana Epa, Platonic 2-groups, 2010 (pdf)

• Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, Higher Structures 1(1):122-146, 2017 (arXiv:1605.09192, hs:30)

In relation to the Monster group:

• Nora Ganter, Subtle Symmetries and the Refined Monster, project description (2018) [pdf, pdf]

• Nora Ganter, Looking for a Refined Monster [arXiv:2405.16410]