special linear group SL(n,C)
E595247
Lie group
complex Lie group
connected Lie group
linear algebraic group
matrix group
simple Lie group
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| SL(n,ℂ) | 3 |
| SL(3,ℂ) | 2 |
| special linear group SL(n,C) canonical | 1 |
| special linear group SL(n,ℂ) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6456479 — 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,C) Context triple: [Lie group, hasExample, special linear group SL(n,C)]
-
A.
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.
-
B.
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.
-
C.
general linear group GL(n,R)
The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.
-
D.
special unitary group SU(n)
The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.
-
E.
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.
- 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,C) Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
general linear group GL(n,R)
The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.
-
D.
special unitary group SU(n)
The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
complex Lie group ⓘ connected Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ simple Lie group ⓘ |
| actsOn | ℂⁿ by the defining representation ⓘ |
| definedAs |
{A ∈ Mₙ(ℂ) | det(A) = 1}
ⓘ
{X ∈ Mₙ(ℂ) | tr(X) = 0} ⓘ |
| hasBorelSubgroups | upper triangular matrices with determinant 1 ⓘ |
| hasCenter | {ζ Iₙ | ζⁿ = 1} ⓘ |
| hasCenterIsomorphicTo | μₙ (group of n-th roots of unity) ⓘ |
| hasConditionOnElements | determinant equal to 1 ⓘ |
| hasDeterminantMapKernelOf | det : GL(n,ℂ) → ℂ* ⓘ |
| hasDimension |
n² − 1 (as complex Lie group)
ⓘ
n² − 1 (as complex vector space) ⓘ |
| hasElementType | n×n complex matrices ⓘ |
| hasFundamentalGroup | 0 (trivial) for n ≥ 2 ⓘ |
| hasIdentityElement | identity matrix Iₙ ⓘ |
| hasLieAlgebra | sl(n,ℂ) ⓘ |
| hasLieAlgebraCondition | trace zero matrices ⓘ |
| hasMaximalTorus | diagonal matrices with determinant 1 ⓘ |
| hasParabolicSubgroups | block upper triangular determinant 1 matrices ⓘ |
| hasQuotient | GL(n,ℂ)/SL(n,ℂ) ≅ ℂ× ⓘ |
| hasRank | n − 1 ⓘ |
| hasRealDimension | 2(n² − 1) ⓘ |
| hasRootSystem | type A_{n−1} ⓘ |
| hasStandardRepresentation | n-dimensional complex representation on ℂⁿ ⓘ |
| hasUniversalCover | itself for n ≥ 2 ⓘ |
| hasWeylGroup |
symmetric group Sₙ
ⓘ
linked to:
Weyl group
|
| isAlgebraicGroupDefinedOver | ℂ ⓘ |
| isCenterNontrivial | true ⓘ |
| isConnected | true ⓘ |
| isDerivedSubgroupOf |
GL(n,ℂ)
ⓘ
linked to:
general linear group GL(n,C)
|
| isFundamentalIn |
algebraic geometry
ⓘ
differential geometry ⓘ representation theory ⓘ theoretical physics ⓘ |
| isKernelOf | determinant homomorphism GL(n,ℂ) → ℂ× ⓘ |
| isNormalSubgroupOf |
GL(n,ℂ)
ⓘ
linked to:
general linear group GL(n,C)
|
| isPerfectGroup | true ⓘ |
| isSimplyConnected | true for n ≥ 2 ⓘ |
| isSimplyLaced | true ⓘ |
| isSubsetOf |
GL(n,ℂ)
ⓘ
linked to:
general linear group GL(n,C)
|
| isUsedIn |
conformal field theory
ⓘ
gauge theory ⓘ quantum field theory ⓘ |
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,C) Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: special linear group SL(n,C)
linked to: special linear group SL(n,C)
subject linked to:
semisimple Lie groups
linked to: special linear group SL(n,C)
subject linked to:
special unitary group SU(n)
linked to: special linear group SL(n,C)
subject linked to:
general linear group GL(n,C)
linked to: special linear group SL(n,C)
subject linked to:
general linear group GL(n,C)
linked to: special linear group SL(n,C)