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.
Observed surface forms (3)
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 NERFINISHED ⓘ |
| isAlgebraicGroupDefinedOver | ℂ ⓘ |
| isCenterNontrivial | true ⓘ |
| isConnected | true ⓘ |
| isDerivedSubgroupOf | GL(n,ℂ) NERFINISHED ⓘ |
| isFundamentalIn |
algebraic geometry
ⓘ
differential geometry ⓘ representation theory ⓘ theoretical physics ⓘ |
| isKernelOf | determinant homomorphism GL(n,ℂ) → ℂ× ⓘ |
| isNormalSubgroupOf | GL(n,ℂ) NERFINISHED ⓘ |
| isPerfectGroup | true ⓘ |
| isSimplyConnected | true for n ≥ 2 ⓘ |
| isSimplyLaced | true ⓘ |
| isSubsetOf | GL(n,ℂ) NERFINISHED ⓘ |
| isUsedIn |
conformal field theory
ⓘ
gauge theory ⓘ quantum field theory ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
SL(3,ℂ)
this entity surface form:
SL(3,ℂ)