|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
complex vector space of dimension n
|
|
gptkbp:centralTo
|
scalar matrices with modulus 1
|
|
gptkbp:compact
|
true
|
|
gptkbp:compactLieGroup
|
true
|
|
gptkbp:connects
|
true
|
|
gptkbp:containsElement
|
n x n unitary matrices
|
|
gptkbp:definedIn
|
complex numbers
|
|
gptkbp:determinant
|
U(n) → U(1)
|
|
gptkbp:dimensions
|
n^2
|
|
gptkbp:fundamentalGroup
|
Z
|
|
gptkbp:hasSpecialCase
|
U(1) is circle group
U(2) is group of 2x2 unitary matrices
|
|
gptkbp:hasSubgroup
|
gptkb:special_unitary_group_SU(n)
|
|
gptkbp:homogeneousSpace
|
U(n)/SU(n) ≅ U(1)
|
|
gptkbp:homotopyGroup
|
π_1(U(n)) = Z
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
unitary group U(n)
|
|
gptkbp:identityElement
|
identity matrix
conjugate transpose
|
|
gptkbp:isQuotientOf
|
U(n)/U(1) ≅ SU(n)/Z_n
|
|
gptkbp:isSimple
|
n=1 is abelian
|
|
gptkbp:Lie_algebra
|
u(n)
|
|
gptkbp:LieAlgebraDimension
|
n^2
|
|
gptkbp:matrixCondition
|
U*U† = I
U† = U inverse
U†U = I
|
|
gptkbp:maximalCompactSubgroupOf
|
gptkb:general_linear_group_GL(n,C)
|
|
gptkbp:maximalTorus
|
diagonal unitary matrices
|
|
gptkbp:notation
|
U(n)
|
|
gptkbp:order
|
infinite
|
|
gptkbp:preserves
|
Hermitian inner product
|
|
gptkbp:realDimension
|
n^2
|
|
gptkbp:realForm
|
gptkb:GL(n,C)
|
|
gptkbp:relatedGroup
|
matrix multiplication
|
|
gptkbp:relatedTo
|
gptkb:general_linear_group_GL(n,C)
gptkb:projective_unitary_group_PU(n)
gptkb:special_linear_group_SL(n,C)
gptkb:special_orthogonal_group_SO(n)
gptkb:symplectic_group_Sp(n)
gptkb:orthogonal_group_O(n)
|
|
gptkbp:type
|
classical group
|
|
gptkbp:usedIn
|
differential geometry
particle physics
quantum mechanics
representation theory
|
|
gptkbp:bfsParent
|
gptkb:Lie_group
|
|
gptkbp:bfsLayer
|
5
|