group theory

∞-Lie theory

# Contents

## Definition

For $n \in \mathbb{N}$ 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 $n \times n$ 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 $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$ (“stable range”) are

$\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.$

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$$\pi_9$
$SO(2)$$\mathbb{Z}$00000000
$SO(3)$$\mathbb{Z}_2$0$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{12}$$\mathbb{Z}_{2}$$\mathbb{Z}_{2}$$\mathbb{Z}_{3}$
$SO(4)$$\mathbb{Z}_{2}$0$\mathbb{Z} \oplus \mathbb{Z}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{12} \oplus \mathbb{Z}_{12}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{3} \oplus \mathbb{Z}_{3}$
$SO(5)$$\mathbb{Z}_2$0$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{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

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

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

Given a manifold $X$, lifts of the structure map $X \to \mathcal{B}O(n)$ of the $O(n)$-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$\mathrm{O}(n)$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)$$\,$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$$\,$$\,$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$

## References

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

The homotopy groups of $O(n)$ 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)