Math 2.0

Topological groups, Lie groups Classical Lie groups Example

SL(2,C)

Builds on Special linear groupRiemann sphereMöbius transformationSU(2)Hermitian matrix

Example

The group SL(2,ℂ)SL(2, \C) is the special linear group of complex 2×22 \times 2 matrices with determinant 11: SL(2,ℂ)={(abcd):a,b,c,d∈ℂ, ad−bc=1}.SL(2, \C) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \C, \ ad - bc = 1}. It is cut out of M2(ℂ)=ℂ4M_2(\C) = \C^4 by the single polynomial equation ad−bc=1ad - bc = 1, whose differential is nonzero at every solution. So it is a complex Lie group of complex dimension 33, that is, of real dimension 66.

It acts on the Riemann sphere ℂ∪{∞}\C \cup \set{\infty} by Möbius transformations z↦(az+b)/(cz+d)z \mapsto (az + b)/(cz + d). Check: the matrix gg acts as the identity map exactly when (az+b)/(cz+d)=z(az + b)/(cz + d) = z for all zz, which forces b=c=0b = c = 0 and a=da = d, and then ad=1ad = 1 gives g=±Ig = \pm I. So the group of Möbius transformations is PSL(2,ℂ)=SL(2,ℂ)/{±I}PSL(2, \C) = SL(2, \C)/\set{\pm I}.

Remarks

Every g∈SL(2,ℂ)g \in SL(2, \C) has a unique polar decomposition g=upg = u p with u∈u \in SU(2) and pp a positive definite Hermitian matrix of determinant 11. These pp form a space homeomorphic to ℝ3\R^3, so SL(2,ℂ)SL(2, \C) is homeomorphic to S3×ℝ3S^3 \times \R^3. Hence it is connected and simply connected, and SU(2)SU(2) is a maximal compact subgroup. Its Lie algebra 𝔰𝔩(2,ℂ)\mathfrak{sl}(2, \C) is the complex simple Lie algebra of type A1A_1. SL(2,ℂ)SL(2, \C) acts on the Hermitian 2×22 \times 2 matrices by X↦gXg∗X \mapsto g X g^*, preserving det⁡X\det X, which is a quadratic form of signature (1,3)(1, 3). This gives a double cover of the identity component of the Lorentz group, so SL(2,ℂ)SL(2, \C) is the spin group Spin(1,3)\mathrm{Spin}(1, 3).

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptReal projective space, object, no evidence levelReal projective spaceobjectRiemann sphere, concept, no evidence levelRiemann sphereconceptVector 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 transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial 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)exampleSL(2,C), example, no evidence levelSL(2,C)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. General linear group object
  4. Group concept
  5. Hermitian matrix concept
  6. Identity matrix object
  7. Lie group concept
  8. Matrix definition
  9. Möbius transformation concept
  10. Real projective space object
  11. Riemann sphere concept
  12. Vector space definition
  13. Ring definition
  14. Square matrix definition
  15. Transpose definition
  16. Linear map definition
  17. Unit of a ring definition
  18. Conjugate transpose definition
  19. Multilinear map definition
  20. Orthogonal matrix definition
  21. Division ring definition
  22. Alternating multilinear map definition
  23. Orthogonal group object
  24. Unitary matrix definition
  25. The quaternions example
  26. Determinant definition
  27. Unitary group object
  28. Special linear group object
  29. Special orthogonal group object
  30. Special unitary group SU(n) example
  31. Rotation group SO(3) example
  32. SU(2) example
  33. SL(2,C) example

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Special_linear_group
About this page
TagJ58ED1GV6N
URIm2://J58ED1GV6N
Kindexample
WritingW0 stub
MSC22E10, 22E46, 20G20
Instance ofLie group
UsesSpecial linear group, Riemann sphere, Möbius transformation, SU(2), Hermitian matrix
Changed2 November 2026

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