orthogonal group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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).

$G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ |
---|---|---|---|---|---|---|---|---|---|

$SO(2)$ | $\mathbb{Z}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$SO(3)$ | $\mathbb{Z}_2$ | 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 | $\mathbb{Z}$ | 0 | 0 |

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**

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 | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |

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

- Paul Garrett,
*Sporadic isogenies to orthogonal groups*, July 2013 (pdf)

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)