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.

The analog for complex Hilbert spaces is the unitary group.

## 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}}$
Revised on October 15, 2011 02:46:40 by Urs Schreiber (82.113.99.46)