special linear group SL(n,R)
E593510
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| SL(n,ℝ) | 3 |
| special linear group SL(n,R) canonical | 1 |
| special linear group SL(n,ℝ) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6456478 — 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: special linear group SL(n,R) Context triple: [Lie group, hasExample, special linear group SL(n,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.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
C.
affine group of R^n
The affine group of ℝⁿ is the group of all invertible affine transformations of n-dimensional real space, combining linear transformations with translations.
-
D.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
E.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
- 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: special linear group SL(n,R) Target entity description: 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.
-
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.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
C.
affine group of R^n
The affine group of ℝⁿ is the group of all invertible affine transformations of n-dimensional real space, combining linear transformations with translations.
-
D.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
E.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Lie algebra
ⓘ
Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ |
| actsOn |
upper half-plane
ⓘ
ℝ^n ⓘ |
| appearsIn |
algebraic groups
ⓘ
differential geometry ⓘ representation theory ⓘ theory of Lie groups ⓘ |
| consistsOf | n×n real matrices with determinant 1 ⓘ |
| definedAs |
{A ∈ M_n(ℝ) | det(A) = 1}
ⓘ
{X ∈ M_n(ℝ) | tr(X) = 0} ⓘ |
| hasCenter |
{I} for n odd
ⓘ
{±I} for n even ⓘ |
| hasDimension |
3
ⓘ
n^2 − 1 ⓘ n^2 − 1 ⓘ |
| hasFundamentalGroup | ℤ for n = 2 ⓘ |
| hasIdentityElement | I_n ⓘ |
| hasLieAlgebra | sl(n,ℝ) ⓘ |
| hasMaximalCompactSubgroup |
SO(2)
ⓘ
linked to:
special orthogonal group SO(n)
SO(n) ⓘ
linked to:
special orthogonal group SO(n)
|
| hasOperation | matrix multiplication ⓘ |
| hasRealRank | n − 1 ⓘ |
| hasRealStructureConstants | true ⓘ |
| isClosedUnder |
matrix multiplication
ⓘ
taking inverses ⓘ |
| isConnected | true for n ≥ 2 ⓘ |
| isDefinedByEquation | det(A) = 1 in M_n(ℝ) ⓘ |
| isDeterminantOnePartOf |
GL(n,ℝ)
ⓘ
linked to:
general linear group GL(n,R)
|
| isLinear | true ⓘ |
| isNonCompact | true for n ≥ 2 ⓘ |
| isNormalSubgroupOf |
GL(n,ℝ)
ⓘ
linked to:
general linear group GL(n,R)
|
| isPathConnected | true for n ≥ 2 ⓘ |
| isPerfectGroup | true for n ≥ 2 ⓘ |
| isRealPointsOf | algebraic group SL_n over ℝ ⓘ |
| isSemisimple | true ⓘ |
| isSimpleAsLieGroup | true for n ≥ 2 except low-dimensional isomorphisms ⓘ |
| isSimplyConnected | false for n ≥ 2 ⓘ |
| isSpecialCaseOf |
SL(n,ℝ)
ⓘ
linked to:
special linear group SL(n,R)
|
| isSubgroupOf |
GL(n,ℝ)
ⓘ
linked to:
general linear group GL(n,R)
|
| isSubsetOf |
GL(n,ℝ)
ⓘ
linked to:
general linear group GL(n,R)
|
| isUnimodular | true ⓘ |
| kernelOf | det : GL(n,ℝ) → ℝ^× ⓘ |
| preserves |
Lebesgue measure on ℝ^n up to normalization
ⓘ
volume on ℝ^n ⓘ |
| quotientBy | {±I} is PSL(2,ℝ) ⓘ |
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: special linear group SL(n,R) Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
semisimple Lie groups
linked to: special linear group SL(n,R)
subject linked to:
special orthogonal group SO(n)
linked to: special linear group SL(n,R)
subject linked to:
general linear group GL(n,R)
linked to: special linear group SL(n,R)
subject linked to:
special linear group SL(n,R)
linked to: special linear group SL(n,R)