group theory

∞-Lie theory

# Contents

## Definition

For $n\in ℕ$ the orthogonal group is the group of isometries of a real $n$-dimensional Hilbert space. This is naturally a Lie group. This is canonically isomorphic to the group of $n×n$ orthogonal matrices.

More generally there is a notion of orthogonal group of an inner product space.

The analog for complex Hilbert spaces is the unitary group.

## Properties

### Homotopy groups

The homotopy groups of $O\left(n\right)$ are for $k\in ℕ$ and for sufficiently large $n$ (“stable range”) are

$\begin{array}{cc}{\pi }_{8k+0}\left(O\right)& ={ℤ}_{2}\\ {\pi }_{8k+1}\left(O\right)& ={ℤ}_{2}\\ {\pi }_{8k+2}\left(O\right)& =0\\ {\pi }_{8k+3}\left(O\right)& =ℤ\\ {\pi }_{8k+4}\left(O\right)& =0\\ {\pi }_{8k+5}\left(O\right)& =0\\ {\pi }_{8k+6}\left(O\right)& =0\\ {\pi }_{8k+7}\left(O\right)& =ℤ\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

In the unstable range for low $n$ they instead start out as follows (e.g. Abanov 09, A.1.1.3.2).

$G$${\pi }_{1}$${\pi }_{2}$${\pi }_{3}$${\pi }_{4}$${\pi }_{5}$${\pi }_{6}$${\pi }_{7}$${\pi }_{8}$${\pi }_{9}$
$\mathrm{SO}\left(2\right)$$ℤ$00000000
$\mathrm{SO}\left(3\right)$${ℤ}_{2}$0$ℤ$${ℤ}_{2}$${ℤ}_{2}$${ℤ}_{12}$${ℤ}_{2}$${ℤ}_{2}$${ℤ}_{3}$
$\mathrm{SO}\left(4\right)$${ℤ}_{2}$0$ℤ\oplus ℤ$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{12}\oplus {ℤ}_{12}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{3}\oplus {ℤ}_{3}$
$\mathrm{SO}\left(5\right)$${ℤ}_{2}$0$ℤ$${ℤ}_{2}$${ℤ}_{2}$0$ℤ$00

### Whitehead tower and higher orientation structures

The Whitehead tower of the orthogonal group plays an important role in applications related to quantum physics.

The first steps are

$\cdots \to \mathrm{Fivebrane}\left(n\right)\to \mathrm{String}\left(n\right)\to \mathrm{Spin}\left(n\right)\to \mathrm{SO}\left(n\right)\to \mathrm{O}\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

Fivebrane group to String group to Spin group to special orthogonal group to orthogonal group.

Given a manifold $X$, lifts of the structure map $X\to ℬO\left(n\right)$ of the $O\left(n\right)$-principal bundle to which the tangent bundle is associated through this tower define, respectively

on $X$.

$\cdots \to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}\left(n\right)$Pin group$\mathrm{Pin}\left(n\right)$Tring group$\mathrm{Tring}\left(n\right)$
special orthogonal group$\mathrm{SO}\left(n\right)$Spin group$\mathrm{Spin}\left(n\right)$String group$\mathrm{String}\left(n\right)$
Lorentz group$\mathrm{O}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
anti de Sitter group$\mathrm{O}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
Narain group$O\left(n,n\right)$
Poincaré group$\mathrm{ISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
super Poincaré group$\mathrm{sISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$

## References

Examples of sporadic (exceptional) isogenies from spin groups onto orthogonal groups are discussed in

The homotopy groups of $O\left(n\right)$ are listed for instance in

• Alexander Abanov, Homotopy groups of Lie groups 2009 (pdf)

Revised on November 4, 2013 01:38:14 by Urs Schreiber (89.204.154.47)