group theory

∞-Lie theory

# Contents

## Definition

The special orthogonal group $\mathrm{SO}\left(n\right)$ is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group $O\left(n\right)$.

For instance for $n=2$ we have $\mathrm{SO}\left(2\right)$ the circle group.

It is the first step in the Whitehead tower of $O\left(n\right)$

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

## Orientation of manifolds

For $X$ an $n$-dimensional manifold a lift of the classifying map $X\to ℬO\left(n\right)$ of the $O\left(n\right)$-principal bundle to which the tangent bundle $TX$ is associated is the same as a choice of orientation of $X$.

$\cdots \to$ 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 02:47:29 by Urs Schreiber (82.113.99.46)