special orthogonal group


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The special orthogonal group or rotation group, denoted SO(n)SO(n), is the group of rotations in a Cartesian space of dimension nn.

This is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group O(n)O(n).

For instance for n=2n=2 we have SO(2)SO(2) the circle group.

It is the first step in the Whitehead tower of O(n)O(n)

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) \,,

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

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

Relation to orientation of manifolds

For XX an nn-dimensional manifold a lift of the classifying map XO(n)X \to \mathcal{B}O(n) of the O(n)O(n)-principal bundle to which the tangent bundle TXT X is associated is the same as a choice of orientation of XX.


For the almost degenerate case n=2n = 2 there are exceptional isomorphisms of Lie groups

SO(2)U(1)Spin(2) 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.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group


For general references see also at orthogonal group.

  • Jim Stasheff, The topology and algebra of SO(n1)SO(n)S n1SO(n-1) \to SO(n) \to S^{n-1}, Herman’s seminar July 2013 (pdf slides)

Revised on January 10, 2017 16:20:59 by Urs Schreiber (