special orthogonal group
Cohomology and Extensions
∞-Lie theory (higher geometry)
Formal Lie groupoids
The special orthogonal group or rotation group, denoted , is the group of rotations in a Cartesian space of dimension .
This is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group .
For instance for we have the circle group.
It is the first step in the Whitehead tower of
the next step of which is the spin group.
In physics the rotation group is related to angular momentum.
As part of the ADE pattern
Relation to orientation of manifolds
For an -dimensional manifold a lift of the classifying map of the -principal bundle to which the tangent bundle is associated is the same as a choice of orientation of .
For the almost degenerate case there are exceptional isomorphisms of Lie groups
with the circle group and spin group in dimension 2.
Fivebrane group string group spin group special orthogonal group orthogonal group.
|group||symbol||universal cover||symbol||higher cover||symbol|
|orthogonal group||Pin group||Tring group|
|special orthogonal group||Spin group||String group|
|anti de Sitter group|
|Poincaré group||Poincaré spin group|
|super Poincaré group|
For general references see also at orthogonal group.
- Jim Stasheff, The topology and algebra of , Herman’s seminar July 2013 (pdf slides)
Revised on January 10, 2017 16:20:59
by Urs Schreiber