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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

Dynkin diagram	Platonic solid	finite subgroup of $SO(3)$	finite subgroup of $SU(2)$	simple Lie group
$A_l$		cyclic group	cyclic group	special unitary group
$D_l$	dihedron/hosohedron	dihedral group	binary dihedral group	special orthogonal group
$E_6$	tetrahedron	tetrahedral group	binary tetrahedral group	E6
$E_7$	cube/octahedron	octahedral group	binary octahedral group	E7
$E_8$	dodecahedron/icosahedron	icosahedral group	binary icosahedral group	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.

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

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)