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gptkbp:instanceOf
|
gptkb:group_of_people
|
|
gptkbp:alsoKnownAs
|
gptkb:U(1)
unitary group of degree 1
one-dimensional torus
|
|
gptkbp:application
|
gptkb:gauge_theory
Fourier analysis
electromagnetism
quantum mechanics
|
|
gptkbp:category
|
gptkb:geometry
gptkb:mathematics
gptkb:topology
gptkb:Lie_groups
group theory
harmonic analysis
physics
representation theory
|
|
gptkbp:centralTo
|
itself
|
|
gptkbp:commutative
|
true
|
|
gptkbp:compact
|
compact
|
|
gptkbp:connects
|
true
|
|
gptkbp:containsElement
|
complex numbers of unit modulus
e^{iθ} for θ in [0, 2π)
|
|
gptkbp:dimensions
|
1
|
|
gptkbp:fundamentalGroup
|
Z (integers)
|
|
gptkbp:generation
|
i (imaginary unit)
|
|
gptkbp:heldBy
|
gptkb:topology
gptkb:Lie_group
abelian group
compact group
subgroup of complex numbers under multiplication
|
|
gptkbp:homomorphismTo
|
real numbers mod 2π
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Circle group
|
|
gptkbp:identityElement
|
1
complex conjugate
|
|
gptkbp:isomorphicTo
|
gptkb:SO(2)
gptkb:unit_circle_in_complex_plane
real numbers mod integers (R/Z)
rotation group in 2D
|
|
gptkbp:order
|
infinite
|
|
gptkbp:parameter
|
angle θ
|
|
gptkbp:representationTheory
|
characters are e^{inθ}, n in Z
|
|
gptkbp:simplyConnected
|
false
|
|
gptkbp:structure
|
gptkb:algebraic_geometry
gptkb:topology
gptkb:Riemannian_manifold
gptkb:Lie_group
group under multiplication
|
|
gptkbp:universalCover
|
real line
|
|
gptkbp:bfsParent
|
gptkb:Topological_Groups
|
|
gptkbp:bfsLayer
|
6
|