Math 2.0

Topological groups, Lie groups Classical Lie groups Example

SL(2,R)

Builds on Special linear groupUpper half-planeMöbius transformationSpecial orthogonal group

Example

The group SL(2,ℝ)SL(2, \R) is the special linear group of real 2×22 \times 2 matrices with determinant 11: SL(2,ℝ)={(abcd):a,b,c,d∈ℝ, ad−bc=1}.SL(2, \R) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \R, \ ad - bc = 1}. It is a closed subgroup of GL2(ℝ)GL_2(\R) cut out by one equation, so it is a Lie group of dimension 33.

It acts on the upper half-plane ℍ={z∈ℂ:Im⁡z>0}\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0} by Möbius transformations z↦az+bcz+d.z \mapsto \frac{az + b}{cz + d}. Check that ℍ\mathbb{H} is preserved: multiplying numerator and denominator by czˉ+dc\bar{z} + d gives Im⁡az+bcz+d=(ad−bc)Im⁡z∣cz+d∣2=Im⁡z∣cz+d∣2>0.\operatorname{Im} \frac{az + b}{cz + d} = \frac{(ad - bc) \operatorname{Im} z}{\abs{cz + d}^2} = \frac{\operatorname{Im} z}{\abs{cz + d}^2} > 0.

Remarks

The matrices II and −I-I act as the identity map, and the quotient PSL(2,ℝ)=SL(2,ℝ)/{±I}PSL(2, \R) = SL(2, \R)/\set{\pm I} is the group of orientation-preserving isometries of the hyperbolic plane in the upper half-plane model. SL(2,ℝ)SL(2, \R) is connected but not compact. Its maximal compact subgroup is the rotation group SO(2), and SL(2,ℝ)SL(2, \R) is homeomorphic to SO(2)×ℝ2SO(2) \times \R^2. So its fundamental group is ℤ\Z, and it is not simply connected. Its universal cover is a Lie group with no faithful finite-dimensional representation. An element g≠±Ig \ne \pm I is called elliptic, parabolic or hyperbolic when ∣tr⁡g∣\abs{\tr g} is less than, equal to, or greater than 22. Its Lie algebra 𝔰𝔩(2,ℝ)\mathfrak{sl}(2, \R) of trace-zero real 2×22 \times 2 matrices is simple, so SL(2,ℝ)SL(2, \R) is a simple Lie group.

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Complex numbers, object, no evidence levelComplex numbersobjectGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptVector 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…objectSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectSL(2,R), example, no evidence levelSL(2,R)example
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  1. Complex numbers object
  2. General linear group object
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  4. Matrix definition
  5. Möbius transformation concept
  6. Vector space definition
  7. Square matrix definition
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  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. Special linear group object
  17. Upper half-plane object
  18. SL(2,R) example

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/SL2(R)
About this page
TagQ0GBCB9RA4
URIm2://Q0GBCB9RA4
Kindexample
WritingW0 stub
MSC22E46, 22E15, 20G20
Instance ofLie group
UsesSpecial linear group, Upper half-plane, Möbius transformation, Special orthogonal group
Same as
Changed2 November 2026

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