SL(2,R)
E683056
Lie group
connected Lie group
linear algebraic group
matrix group
semisimple Lie group
simple Lie group
SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
connected Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ semisimple Lie group ⓘ simple Lie group ⓘ |
| actionType | fractional linear transformations ⓘ |
| actsByIsometriesOn | hyperbolic plane ⓘ |
| actsOn | upper half-plane ⓘ |
| CartanDecomposition | KAK decomposition with K=SO(2) ⓘ |
| center | {±I} ⓘ |
| connected | true ⓘ |
| definedAs | set of 2×2 real matrices with determinant 1 ⓘ |
| determinantCondition | det(A)=1 ⓘ |
| dimension | 3 ⓘ |
| fullName | special linear group of 2×2 real matrices ⓘ |
| fundamentalGroup | Z ⓘ |
| groupOperation | matrix multiplication ⓘ |
| hasDiscreteSubgroup | SL(2,Z) NERFINISHED ⓘ |
| identityElement | 2×2 identity matrix ⓘ |
| isomorphicTo | SU(1,1) as real Lie groups ⓘ |
| IwasawaDecomposition | KAN decomposition with K=SO(2) ⓘ |
| LieAlgebra | sl(2,R) ⓘ |
| LieAlgebraDefinedAs | 2×2 real matrices with trace 0 ⓘ |
| LieAlgebraDimension | 3 ⓘ |
| maximalCompactSubgroup | SO(2) NERFINISHED ⓘ |
| nonCompact | true ⓘ |
| overField | R ⓘ |
| quotientByCenter | PSL(2,R) NERFINISHED ⓘ |
| rank | 1 ⓘ |
| realFormOf | SL(2,C) NERFINISHED ⓘ |
| relatedGroup |
PSL(2,R)
NERFINISHED
ⓘ
SU(1,1) NERFINISHED ⓘ |
| simplyConnected | false ⓘ |
| symbol | SL(2,ℝ) NERFINISHED ⓘ |
| topology | diffeomorphic to S^1×R^2 ⓘ |
| universalCover | universal covering group of SL(2,R) ⓘ |
| usedIn |
automorphic forms
ⓘ
conformal field theory ⓘ harmonic analysis ⓘ hyperbolic geometry ⓘ mathematical physics ⓘ number theory ⓘ representation theory ⓘ theory of differential equations ⓘ theory of modular forms ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.