#
nLab

special orthogonal group

### Context

#### Group Theory

**group theory**

### Classical groups

### Finite groups

### Group schemes

### Topological groups

### Lie groups

### Super-Lie groups

### Higher groups

### Cohomology and Extensions

#### $\infty$-Lie theory

**∞-Lie theory**

## Background

### Smooth structure

### Higher groupoids

### Lie theory

## ∞-Lie groupoids

## ∞-Lie algebroids

## Cohomology

## Homotopy

## Examples

### $\infty$-Lie groupoids

### $\infty$-Lie groups

### $\infty$-Lie algebroids

### $\infty$-Lie algebras

# 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

### 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.

group | symbol | universal cover | symbol | higher cover | symbol |
---|

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

For general references see also at *orthogonal group*.

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

Revised on October 4, 2013 01:15:52
by

Urs Schreiber
(82.169.114.243)