sl(2,C)
E683055
complex Lie algebra
complex vector space
matrix Lie algebra
semisimple Lie algebra
simple Lie algebra
sl(2,C) is the three-dimensional complex Lie algebra of 2×2 complex traceless matrices, fundamental in representation theory and the study of Lie groups and algebras.
All labels observed (1)
| Label | Occurrences |
|---|---|
| sl(2,C) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7705308 — 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,C) Context triple: [SL(2,C), hasLieAlgebra, sl(2,C)]
-
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.
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B.
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.
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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.
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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.
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E.
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.
- 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,C) Target entity description: sl(2,C) is the three-dimensional complex Lie algebra of 2×2 complex traceless matrices, fundamental in representation theory and the study of Lie groups and algebras.
-
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 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.
-
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.
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E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex Lie algebra
ⓘ
complex vector space ⓘ matrix Lie algebra ⓘ semisimple Lie algebra ⓘ simple Lie algebra ⓘ |
| allFiniteDimensionalRepresentations | completely reducible ⓘ |
| appearsIn |
representation theory of semisimple Lie algebras
ⓘ
theory of angular momentum in quantum mechanics ⓘ theory of special relativity via SL(2,C) ⓘ |
| associatedLieGroup | SL(2,C) ⓘ |
| automorphismGroup | inner automorphisms induced by SL(2,C) ⓘ |
| BorelSubalgebra | upper triangular traceless matrices ⓘ |
| bracketOperation | matrix commutator ⓘ |
| CartanSubalgebraDimension | 1 ⓘ |
| CasimirElement | central element in U(sl(2,C)) ⓘ |
| center | {0} ⓘ |
| consistsOf | 2×2 complex matrices with trace 0 ⓘ |
| definition | Lie algebra of 2×2 complex traceless matrices ⓘ |
| dimension | 3 over C ⓘ |
| field | C ⓘ |
| finiteDimensionalIrrepsClassifiedBy | highest weight (nonnegative integer) ⓘ |
| hasAdjointRepresentation | 3-dimensional irreducible representation ⓘ |
| hasBasisElement |
E = [[0,1],[0,0]]
ⓘ
F = [[0,0],[1,0]] ⓘ H = [[1,0],[0,-1]] ⓘ |
| hasFundamentalRepresentation | 2-dimensional defining representation on C^2 ⓘ |
| hasIrreducibleRepresentation | (n+1)-dimensional representation for each n ∈ N ⓘ |
| hasRootSpaceDecomposition | H ⊕ CE ⊕ CF ⓘ |
| isComplexificationOf |
sl(2,R)
ⓘ
linked to:
sl(2,ℝ)
su(2) ⓘ |
| isIsomorphicTo | so(3,C) ⓘ |
| isPrototypeFor | sl(2)-subalgebras in semisimple Lie algebras ⓘ |
| isSemisimple | true ⓘ |
| isSimple | true ⓘ |
| isSubalgebraOf | gl(2,C) ⓘ |
| KillingForm | nondegenerate ⓘ |
| LieBracketRelation |
[E,F] = H
ⓘ
[H,E] = 2E ⓘ [H,F] = -2F ⓘ |
| notation | sl(2,ℂ) ⓘ |
| rank | 1 ⓘ |
| realForm |
sl(2,R)
ⓘ
linked to:
sl(2,ℝ)
su(2) ⓘ |
| rootSystemType | A1 ⓘ |
| universalCoveringGroup | SL(2,C) ⓘ |
| universalEnvelopingAlgebra | U(sl(2,C)) ⓘ |
| usedAs | basic example in Lie theory textbooks ⓘ |
| WeylGroup | Z2 ⓘ |
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,C) Description of subject: sl(2,C) is the three-dimensional complex Lie algebra of 2×2 complex traceless matrices, fundamental in representation theory and the study of Lie groups and algebras.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.