nLab
orthogonal group

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Definition

For n∈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

⋯→Fivebrane(n)→String(n)→Spin(n)→SO(n)→O(n).\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→ℬO(n) of the O(n)-principal bundle to which the tangent bundle is associated through this tower define, respectively

on X.

⋯→ fivebrane group → string group → spin group → special orthogonal group → orthogonal group

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)Pin groupPin(n)Tring groupTring(n)
special orthogonal groupSO(n)Spin groupSpin(n)String groupString(n)
Lorentz groupO(n,1)Spin(n,1)
anti de Sitter groupO(n,2)Spin(n,2)
Narain groupO(n,n)
Poincaré groupISO(n,1)
super Poincaré groupsISO(n,1)
Revised on October 15, 2011 02:46:40 by Urs Schreiber (82.113.99.46)
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