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.
All labels observed (2)
How this entity was disambiguated
This entity first appeared as the object of triple T7705326 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: SL(2,R) Context triple: [SL(2,C), containsSubgroup, SL(2,R)]
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
PSL(2,ℝ)
PSL(2,ℝ) is the Lie group of orientation-preserving isometries of the hyperbolic plane, realized as 2×2 real matrices with determinant 1 modulo their center.
-
C.
SL(2,ℤ)
SL(2,ℤ) is the group of 2×2 integer matrices with determinant 1, fundamental in number theory, geometry, and the theory of modular forms.
-
D.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: SL(2,R) Target entity description: 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.
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
PSL(2,ℝ)
PSL(2,ℝ) is the Lie group of orientation-preserving isometries of the hyperbolic plane, realized as 2×2 real matrices with determinant 1 modulo their center.
-
C.
SL(2,ℤ)
SL(2,ℤ) is the group of 2×2 integer matrices with determinant 1, fundamental in number theory, geometry, and the theory of modular forms.
-
D.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
E.
special linear group SL(n,C)
The special linear group SL(n,ℂ) is the Lie group of n×n complex matrices with determinant 1, fundamental in representation theory, geometry, and many areas of modern mathematics and physics.
- F. None of above. chosen
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)
ⓘ
linked to:
SL(2,ℤ)
|
| 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)
ⓘ
linked to:
special orthogonal group SO(n)
|
| nonCompact | true ⓘ |
| overField | R ⓘ |
| quotientByCenter |
PSL(2,R)
ⓘ
linked to:
PSL(2,ℝ)
|
| rank | 1 ⓘ |
| realFormOf | SL(2,C) ⓘ |
| relatedGroup |
PSL(2,R)
ⓘ
linked to:
PSL(2,ℝ)
SU(1,1) ⓘ |
| simplyConnected | false ⓘ |
| symbol |
SL(2,ℝ)
ⓘ
linked to:
SL(2,R)
|
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: SL(2,R) Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: SL(2,R)
linked to: SL(2,R)
linked to: SL(2,R)