|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
upper half-plane
projective line
|
|
gptkbp:centralTo
|
{±I}
|
|
gptkbp:complexification
|
gptkb:SL(2,_C)
|
|
gptkbp:dimensions
|
3
|
|
gptkbp:discreteSubgroup
|
gptkb:modular_group_SL(2,_Z)
|
|
gptkbp:field
|
real numbers
|
|
gptkbp:fullName
|
gptkb:Special_Linear_Group_of_2x2_real_matrices
|
|
gptkbp:fundamentalGroup
|
infinite cyclic
|
|
gptkbp:groupOrder
|
infinite
|
|
gptkbp:hasBorelSubgroup
|
upper triangular matrices with determinant 1
|
|
gptkbp:hasCartanDecomposition
|
gptkb:KAK_decomposition
|
|
gptkbp:hasDiscreteSeriesRepresentations
|
yes
|
|
gptkbp:hasIwasawaDecomposition
|
KAN decomposition
|
|
gptkbp:hasMaximalTorus
|
diagonal matrices with determinant 1
|
|
gptkbp:hasPrincipalSeriesRepresentations
|
yes
|
|
gptkbp:hasRealRank
|
1
|
|
gptkbp:hasSubgroup
|
gptkb:SO(2)
gptkb:GL(2,_R)
diagonal matrices with determinant 1
upper triangular matrices with determinant 1
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SL(2, R)
|
|
gptkbp:isomorphicTo
|
group of orientation-preserving isometries of the hyperbolic plane
|
|
gptkbp:Lie_algebra
|
gptkb:sl(2,_R)
|
|
gptkbp:matrixCondition
|
determinant equals 1
|
|
gptkbp:matrixSize
|
2x2
|
|
gptkbp:maximalCompactSubgroup
|
gptkb:SO(2)
|
|
gptkbp:notation
|
gptkb:SL(2,_ℝ)
|
|
gptkbp:rank
|
1
|
|
gptkbp:realForm
|
gptkb:SL(2,_C)
|
|
gptkbp:relatedGroup
|
simple
non-abelian
connected
non-compact
|
|
gptkbp:relatedTo
|
gptkb:SU(2)
gptkb:SL(2,_C)
gptkb:PSL(2,_R)
|
|
gptkbp:type
|
A1
|
|
gptkbp:universalCover
|
universal covering group of SL(2, R)
|
|
gptkbp:usedIn
|
gptkb:hyperbolic_geometry
differential geometry
modular forms
number theory
representation theory
theory of automorphic forms
|
|
gptkbp:Weyl_group
|
gptkb:Z/2Z
|
|
gptkbp:bfsParent
|
gptkb:SL(2,_Z)
gptkb:SL2
|
|
gptkbp:bfsLayer
|
6
|