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Platonic 2-group

Contents

Context

Group Theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

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 a finite subgroup of SU(2).

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

Properties

Relation to the string 2-group

Proposition

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

𝒢 uni[i] BG ADE B 3/|G ADE| B 3i String(SU(2)) BSU(2) c 2 B 3U(1), \array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \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 ii is the canonical inclusion of roots of unity, i:/|G|U(1)i: \mathbb{Z}/{|G|} \hookrightarrow U(1).

(Epa-Ganter 16, prop. 4.1)

References

Last revised on September 9, 2018 at 06:21:30. See the history of this page for a list of all contributions to it.