Math 2.0

Topological groups, Lie groups Classical Lie groups Example

SU(2)

Builds on Special unitary group SU(n)Rotation group SO(3)The quaternions

Example

The group SU(2)SU(2) is the special unitary group for n=2n = 2. Its elements are exactly the matrices U=(a−bˉbaˉ),a,b∈ℂ,∣a∣2+∣b∣2=1.U = \begin{pmatrix} a & -\bar{b} \\ b & \bar{a} \end{pmatrix}, \qquad a, b \in \C, \quad \abs{a}^2 + \abs{b}^2 = 1. Check: for such a matrix, U∗U=(∣a∣2+∣b∣200∣a∣2+∣b∣2)=I2U^* U = \begin{pmatrix} \abs{a}^2 + \abs{b}^2 & 0 \\ 0 & \abs{a}^2 + \abs{b}^2 \end{pmatrix} = I_2 and det⁡U=∣a∣2+∣b∣2=1\det U = \abs{a}^2 + \abs{b}^2 = 1. Conversely, a unitary matrix with determinant 11 has its second column determined by its first column in this way.

So the map U↦(a,b)U \mapsto (a, b) identifies SU(2)SU(2) with the unit sphere S3⊂ℂ2=ℝ4S^3 \subset \C^2 = \R^4. It is a compact Lie group of dimension 33.

Remarks

Under the usual identification of ℂ2\C^2 with the quaternions, SU(2)SU(2) becomes isomorphic to the group of unit quaternions. Since S3S^3 is simply connected, SU(2)SU(2) is simply connected. SU(2)SU(2) acts on the trace-zero Hermitian 2×22 \times 2 matrices, a real vector space of dimension 33, by X↦UXU∗X \mapsto U X U^*. This action preserves −det⁡X-\det X, which is a positive definite quadratic form on that space. It gives a surjective homomorphism SU(2)→SO(3)SU(2) \to SO(3) onto the rotation group with kernel {±I2}\set{\pm I_2}. So SU(2)SU(2) is the universal cover of SO(3)SO(3), and it is also the spin group Spin(3)\mathrm{Spin}(3). Its Lie algebra 𝔰𝔲(2)\mathfrak{su}(2) has basis iσ1,iσ2,iσ3i\sigma_1, i\sigma_2, i\sigma_3, where σ1,σ2,σ3\sigma_1, \sigma_2, \sigma_3 are the Pauli matrices.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptIdentity 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 spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionThe quaternions, example, no evidence levelThe quaternionsexampleUnitary group, object, no evidence levelUnitary groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Determinant concept
  3. Group concept
  4. Identity matrix object
  5. Lie group concept
  6. Matrix definition
  7. Real projective space object
  8. Vector space definition
  9. Ring definition
  10. Square matrix definition
  11. Transpose definition
  12. Linear map definition
  13. Unit of a ring definition
  14. Conjugate transpose definition
  15. Orthogonal matrix definition
  16. Multilinear map definition
  17. Division ring definition
  18. Orthogonal group object
  19. Unitary matrix definition
  20. Alternating multilinear map definition
  21. The quaternions example
  22. Unitary group object
  23. Determinant definition
  24. Special orthogonal group object
  25. Special unitary group SU(n) example
  26. Rotation group SO(3) example
  27. SU(2) example

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Special_unitary_group
About this page
TagE5P8AG6CK7
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Kindexample
WritingW0 stub
MSC22E15, 22C05, 20G20
Instance ofCompact Lie group
UsesSpecial unitary group SU(n), Rotation group SO(3), The quaternions

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