general linear group GL(n,C)
E595246
The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| GL(n,ℂ) | 0 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
complex Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ topological group ⓘ |
| actsOn | ℂⁿ by left multiplication ⓘ |
| appearsIn | classification of complex representations of finite groups via embeddings ⓘ |
| center | {λIₙ : λ ∈ ℂˣ} ⓘ |
| centerIsomorphicTo | ℂˣ ⓘ |
| conditionForMembership | det(A) ≠ 0 ⓘ |
| containsSubgroup |
Borel subgroup of upper triangular invertible matrices
NERFINISHED
ⓘ
SL(n,ℂ) NERFINISHED ⓘ U(n) ⓘ |
| definedAs | group of all invertible n×n complex matrices ⓘ |
| definedByPolynomialCondition | det(A) ≠ 0 ⓘ |
| determinantMap | det : GL(n,ℂ) → ℂˣ ⓘ |
| determinantMapIs | surjective group homomorphism ⓘ |
| dimensionAsComplexLieGroup | n² ⓘ |
| dimensionAsRealManifold | 2n² ⓘ |
| fundamentalGroup | ℤ ⓘ |
| hasDeterminantCharacter | det : GL(n,ℂ) → ℂˣ ⓘ |
| hasMaximalTorus | diagonal invertible matrices ⓘ |
| identityElement | n×n identity matrix ⓘ |
| inverseOperation | matrix inverse ⓘ |
| isAlgebraicGroupOver | ℂ ⓘ |
| isConnected | true ⓘ |
| isNonAbelian | true ⓘ |
| isOpenSubsetOf | Mₙ(ℂ) with respect to standard topology ⓘ |
| isReductive | true ⓘ |
| isSimplyConnected | false ⓘ |
| isSolvable | false ⓘ |
| kernelOfDeterminant | SL(n,ℂ) NERFINISHED ⓘ |
| LieAlgebra | 𝔤𝔩(n,ℂ) ⓘ |
| LieAlgebraDefinedAs | all n×n complex matrices with usual commutator bracket ⓘ |
| maximalCompactSubgroup | U(n) NERFINISHED ⓘ |
| notationVariant | GLₙ(ℂ) NERFINISHED ⓘ |
| overField | ℂ ⓘ |
| parameter | n ∈ ℕ, n ≥ 1 ⓘ |
| quotientByCenter | PGL(n,ℂ) ⓘ |
| rank | n ⓘ |
| roleInMathematics |
basic example in Lie theory
ⓘ
basic example in algebraic geometry ⓘ central in representation theory ⓘ fundamental in linear algebra ⓘ |
| standardRepresentation | action on ℂⁿ ⓘ |
| subgroupDefinedAs | SL(n,ℂ) = {A ∈ GL(n,ℂ) : det(A) = 1} ⓘ |
| underOperation | matrix multiplication ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.