Math 2.0

Topological groups, Lie groups Classical Lie groups Example

Rotation group SO(3)

Builds on Special orthogonal groupCross productReal projective spaceLie group

Example

The rotation group is the special orthogonal group in dimension 33: SO(3)={R∈M3(ℝ):RTR=I3, det⁡R=1}.SO(3) = \set{R \in M_3(\R) : R^T R = I_3, \ \det R = 1}. Each R∈SO(3)R \in SO(3) is a rotation by some angle θ∈[0,π]\theta \in [0, \pi] about some line through the origin. For example, the rotation by θ\theta about the zz-axis is Rz(θ)=(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001).R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}. Check: its columns are orthonormal, so Rz(θ)TRz(θ)=I3R_z(\theta)^T R_z(\theta) = I_3, and its determinant is cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1.

SO(3)SO(3) is a compact Lie group: it is a closed subgroup of GL3(ℝ)GL_3(\R), so a Lie group, and it is closed and bounded in M3(ℝ)M_3(\R).

Remarks

SO(3)SO(3) is connected and has dimension 33. Its Lie algebra 𝔰𝔬(3)\mathfrak{so}(3) is the space of real skew-symmetric 3×33 \times 3 matrices, which is isomorphic to ℝ3\R^3 with the cross product as bracket. As a manifold, SO(3)SO(3) is diffeomorphic to the real projective space ℝP3\R P^3. Its fundamental group has order 22, so it is not simply connected. Its universal cover is SU(2)SU(2), the group of unit quaternions, and the covering map has kernel {±1}\set{\pm 1}.

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Cross product, concept, no evidence levelCross productconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionReal projective space, object, no evidence levelReal projective spaceobjectVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectRotation group SO(3), example, no evidence levelRotation group SO(3)example
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  1. Cross product concept
  2. Identity matrix object
  3. Lie group concept
  4. Matrix definition
  5. Real projective space object
  6. Vector space definition
  7. Square matrix definition
  8. Transpose definition
  9. Linear map definition
  10. Orthogonal matrix definition
  11. Multilinear map definition
  12. Orthogonal group object
  13. Alternating multilinear map definition
  14. Determinant definition
  15. Special orthogonal group object
  16. Rotation group SO(3) example

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Further reading

  1. Web https://en.wikipedia.org/wiki/3D_rotation_group
About this page
Tag263GGJFA21
URIm2://263GGJFA21
Kindexample
WritingW0 stub
MSC22E15, 22C05, 20G20
Instance ofCompact Lie group
UsesSpecial orthogonal group, Cross product, Real projective space, Lie group
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