general linear group GL(n, C)

URI: https://gptkb.org/entity/general_linear_group_GL(n,_C)
GPTKB entity
Statements (48)
Predicate Object
gptkbp:instanceOf gptkb:group_of_people
gptkb:Lie_group
gptkbp:actsOn C^n
gptkbp:centralTo scalar matrices
C* · I
gptkbp:compact true
gptkbp:complexification gptkb:GL(n,_R)
gptkbp:contains gptkb:special_linear_group_SL(n,_C)
gptkb:unitary_group_U(n)
gptkb:orthogonal_group_O(n,_C)
gptkbp:containsElement invertible n x n matrices over C
gptkbp:definedIn complex numbers
gptkbp:determinant det: GL(n, C) → C*
gptkbp:dimensions n^2
gptkbp:hasConnection false
true
gptkbp:hasDeterminantHomomorphism true
gptkbp:hasMaximalSubgroup gptkb:unitary_group_U(n)
https://www.w3.org/2000/01/rdf-schema#label general linear group GL(n, C)
gptkbp:identityElement identity matrix
gptkbp:isAlgebraicGroup C
true
gptkbp:isClassicalGroup true
gptkbp:isConnectedComponentOfIdentity itself
gptkbp:isDenseIn M(n, C)
gptkbp:isFundamentalGroupOf GL(n, C) is Z
gptkbp:isHomogeneousSpaceFor itself
gptkbp:isLieGroupOver C
gptkbp:isMatrixGroup true
gptkbp:isNonAbelian true
gptkbp:isNonDiscrete true
gptkbp:isNonFinite true
gptkbp:isOpenSubsetOf M(n, C)
gptkbp:isParentalGroupOf all n-dimensional representations over C
gptkbp:isQuotientOf gptkb:projective_general_linear_group_PGL(n,_C)
gptkbp:isReductive true
gptkbp:isSimple false
gptkbp:isSmoothManifold true
gptkbp:isTopologicalGroup true
gptkbp:isUniversalGroupFor linear representations of groups
gptkbp:isZariskiOpenIn M(n, C)
gptkbp:Lie_algebra gl(n, C)
gptkbp:notation gptkb:GL(n,_C)
gptkbp:relatedGroup matrix multiplication
gptkbp:specialSubgroup gptkb:special_linear_group_SL(n,_C)
gptkbp:bfsParent gptkb:general_linear_group_GL(n,_R)
gptkb:special_linear_group_SL(n,_C)
gptkbp:bfsLayer 7