|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
gptkb:hyperbolic_plane
upper half-plane
|
|
gptkbp:centralTo
|
{I, -I}
|
|
gptkbp:complexification
|
gptkb:SL(2,C)
|
|
gptkbp:defines
|
group of 2x2 real matrices with determinant 1
|
|
gptkbp:dimensions
|
3
|
|
gptkbp:field
|
real numbers
|
|
gptkbp:fullName
|
gptkb:Special_Linear_Group_of_2x2_Real_Matrices
|
|
gptkbp:fundamentalGroup
|
infinite cyclic group
|
|
gptkbp:hasCartanDecomposition
|
gptkb:KAK_decomposition
|
|
gptkbp:hasComplementarySeriesRepresentations
|
yes
|
|
gptkbp:hasDiscreteSeriesRepresentations
|
yes
|
|
gptkbp:hasDiscreteSubgroup
|
gptkb:Fuchsian_groups
gptkb:modular_group_SL(2,Z)
|
|
gptkbp:hasFiniteDimensionalRepresentations
|
yes
|
|
gptkbp:hasInfiniteDimensionalRepresentations
|
yes
|
|
gptkbp:hasIwasawaDecomposition
|
KAN decomposition
|
|
gptkbp:hasMaximalSubgroup
|
gptkb:SO(2)
|
|
gptkbp:hasPrincipalSeriesRepresentations
|
yes
|
|
gptkbp:hasRealRank
|
1
|
|
gptkbp:hasSubgroup
|
gptkb:SO(2)
diagonal matrices with determinant 1
upper triangular matrices with determinant 1
|
|
gptkbp:hasUnitaryDual
|
infinite dimensional representations
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SL(2,R)
|
|
gptkbp:isomorphicTo
|
gptkb:symplectic_group_Sp(2,R)
|
|
gptkbp:isQuotientOf
|
gptkb:PSL(2,R)
|
|
gptkbp:Lie_algebra
|
gptkb:sl(2,R)
|
|
gptkbp:order
|
infinite
|
|
gptkbp:rank
|
1
|
|
gptkbp:realForm
|
gptkb:SL(2,C)
|
|
gptkbp:relatedGroup
|
simple
non-abelian
connected
non-compact
|
|
gptkbp:relatedTo
|
gptkb:SL(2,C)
gptkb:SU(2)
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
representation theory
theory of automorphic forms
|
|
gptkbp:bfsParent
|
gptkb:SL(2,C)
gptkb:SL(2,R)_geometry
gptkb:PSL(2,R)
gptkb:modular_group_SL(2,Z)
gptkb:SL(2,Z)
gptkb:SL(2,R)_symmetry
|
|
gptkbp:bfsLayer
|
7
|