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.
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) NERFINISHED ⓘ |
| 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)
NERFINISHED
ⓘ
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)
NERFINISHED
ⓘ
su(2) ⓘ |
| rootSystemType | A1 ⓘ |
| universalCoveringGroup | SL(2,C) NERFINISHED ⓘ |
| universalEnvelopingAlgebra | U(sl(2,C)) NERFINISHED ⓘ |
| usedAs | basic example in Lie theory textbooks ⓘ |
| WeylGroup | Z2 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.