group theory

∞-Lie theory

# Contents

## 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

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$simple Lie group
$A_l$cyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupE6
$E_7$cube/octahedronoctahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupE8

### 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.

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

groupsymboluniversal coversymbolhigher coversymbol
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

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