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SO(3)

Contents

Contents

Idea

The special orthogonal group in dimension 3.

Properties

Cohomology

Proposition

The integral cohomology ring of the classifying space BSO(3)B SO(3) is

H (BSO(3),)[p 1,W 3]/(2W 3), H^\bullet\big( B SO(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3) \,,

where

This is a special case of Brown 82, theorem 1.5, which is also reviewed as Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25.


Finite subgroups

Theorem

(ADE classification of finite subgroups of SO(3) and Spin(3)\simeq SU(2))

The finite subgroups of the special orthogonal group SO(3)SO(3) as well as the finite subgroups of the special unitary group SU(2) are, up to conjugation, given by the following classification:

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

Here under the double cover projection (using the exceptional isomorphism SU(2)Spin(3)SU(2) \simeq Spin(3))

SU(2)Spin(3)πSO(3) SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)

all the finite subgroups of SU(2)SU(2) except the odd-order cyclic groups are the preimages of the corresponding finite subgroups of SO(3)SO(3), in that we have pullback diagrams

exp(πi1n) = /(2n) AA Spin(2) AA Spin(3) (pb) (pb) π Ad exp(πi1n) = /n AA SO(2) AA SO(3) \array{ \left\langle \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) \right\rangle & = & \mathbb{Z}/(2n) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) } \right\rangle & = & \mathbb{Z}/n &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

exhibiting the even order cyclic groups as subgroups of Spin(2), including the the minimal case of the group of order 2

exp(πi)=1 = /2 AA Spin(2) AA Spin(3) (pb) (pb) π Ad exp(πi)=e = 1 AA SO(2) AA SO(3) \array{ \left\langle \exp \left( \pi \mathrm{i} \right) = -1 \right\rangle & = & \mathbb{Z}/2 &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \right) } = e \right\rangle & = & 1 &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

as well as

exp(πi1n),j = 2D 2n AA Pin (2) AA Spin(3) (pb) (pb) π Ad exp(πi1n),Ad j D 2n AA O(2) AA SO(3) \array{ \left\langle \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j} \right\rangle &=& 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n} \right) }, \, Ad_{\mathrm{j}} \right\rangle && D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

exhibiting the binary dihedral groups as sitting inside the Pin(2)-subgroup of Spin(3),

but only commuting diagrams

exp(2πi12n+1) = /(2n+1) AA Spin(3) π Ad exp(2πi12n+1) = /(2n+1) AA SO(2) AA SO(3) \array{ \left\langle \exp \left( 2 \pi \mathrm{i} \tfrac{1}{{2n+1}} \right) \right\rangle & = & \mathbb{Z}/(2n+1) &&\overset{\phantom{AA}}{\hookrightarrow}&& Spin(3) \\ && \big\downarrow && && \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( 2 \pi \mathrm{i} \tfrac{1}{2n+1} \right) } \right\rangle & = & \mathbb{Z}/(2n+1) &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

for the odd order cyclic subgroups.

This goes back to (Klein 1884, chapter I). Full proof for SO(3)SO(3) is spelled out for instance in (Rees 05, theorem 11, De Visscher 11). The proof for the case of SL(2,)SL(2,\mathbb{C}) is spelled out in (Miller-Blichfeldt-Dickson 16) reviewed in (Serrano 14, section 2). The proof of the case for SU(2)SU(2) given the result for SO(3)SO(3) is spelled out in Keenan 03, theorem 4.


rotation groups in low dimensions:

sp. orth. groupspin grouppin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)
SO(9)Spin(9)

see also


References

  • Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)

See also

On the integral cohomology of the classifying space:

  • Edgar H. Brown, The Cohomology of BSO nB SO_n and BO nBO_n with Integer Coefficients, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)

reviewed in

  • Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)

Last revised on April 10, 2019 at 11:55:23. See the history of this page for a list of all contributions to it.