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special orthogonal group

Definition

The special orthogonal group or rotation group, denoted $SO(n)$, is the group of rotations in a Cartesian space of dimension $n$.

This is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group $O(n)$.

For instance for $n=2$ we have $SO(2)$ the circle group.

It is the first step in the Whitehead tower of $O(n)$

$$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,,$$

the next step of which is the spin group.

In physics the rotation group is related to angular momentum.

Properties

As part of the ADE pattern

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

Relation to orientation of manifolds

For $X$ an $n$-dimensional manifold a lift of the classifying map $X \to \mathcal{B}O(n)$ of the $O(n)$-principal bundle to which the tangent bundle $T X$ is associated is the same as a choice of orientation of $X$.

Examples

• For the almost degenerate case $n = 2$ there are exceptional isomorphisms of Lie groups

  $$SO(2) \simeq U(1) \simeq Spin(2)$$

  with the circle group and spin group in dimension 2.

• the case of SO(8) is special, since in the ADE classification of simple Lie groups it corresponds to D4, which makes its representation theory enjoy triality.

rotation groups in low dimensions:

sp. orth. group	spin group	pin group
SO(2)	Spin(2)	Pin(2)
SO(3)	Spin(3)
SO(4)	Spin(4)
	Spin(5)
	Spin(6)
	Spin(7)
SO(8)	Spin(8)

see also

• Sp(n).Sp(1)

• finite rotation groups

Related concepts

$\cdots \to$ Fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$Pin(n)$	Tring group	$Tring(n)$
special orthogonal group	$SO(n)$	Spin group	$Spin(n)$	String group	$String(n)$
Lorentz group	$\mathrm{O}(n,1)$	$\,$	$Spin(n,1)$	$\,$	$\,$
anti de Sitter group	$\mathrm{O}(n,2)$	$\,$	$Spin(n,2)$	$\,$	$\,$
conformal group	$\mathrm{O}(n+1,t+1)$	$\,$
Narain group	$O(n,n)$
Poincaré group	$ISO(n,1)$	Poincaré spin group	$\widehat {ISO}(n,1)$	$\,$	$\,$
super Poincaré group	$sISO(n,1)$	$\,$	$\,$	$\,$	$\,$
superconformal group

• Euler angles

References

For general references see also at orthogonal group.

• Jim Stasheff, The topology and algebra of $SO(n-1) \to SO(n) \to S^{n-1}$, Herman’s seminar July 2013 (pdf slides)