Statements (53)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
|
| gptkbp:alsoKnownAs |
gptkb:U(1)
unitary group of degree 1 1-dimensional torus |
| gptkbp:appearsIn |
gptkb:gauge_theory
gptkb:topology Fourier analysis differential geometry electromagnetism harmonic analysis physics quantum mechanics representation theory |
| gptkbp:automorphismGroup |
integers mod 2
|
| gptkbp:centralTo |
itself
|
| gptkbp:commutative |
yes
|
| gptkbp:compact |
compact
|
| gptkbp:connects |
yes
|
| gptkbp:containsElement |
complex numbers of unit modulus
numbers of the form e^{iθ} for θ in [0, 2π) |
| gptkbp:dimensions |
1
|
| gptkbp:first_cohomology_group |
integers
|
| gptkbp:first_homology_group |
integers
|
| gptkbp:fundamentalGroup |
isomorphic to the integers
|
| gptkbp:generation |
e^{iθ} for θ irrational generates dense subgroup
|
| gptkbp:Haar_measure |
Lebesgue measure on [0,2π)
|
| gptkbp:hasElementOrder |
infinite
|
| gptkbp:hasSubgroup |
finite cyclic groups
|
| gptkbp:heldBy |
gptkb:topology
gptkb:Lie_group abelian group compact group subgroup of complex numbers under multiplication |
| gptkbp:homomorphism |
maps to itself by raising to integer powers
|
| https://www.w3.org/2000/01/rdf-schema#label |
circle group
|
| gptkbp:identityElement |
1
|
| gptkbp:Lie_algebra |
real numbers with addition
|
| gptkbp:operator |
complex multiplication
|
| gptkbp:parameter |
angle θ
|
| gptkbp:Pontryagin_dual |
integers
|
| gptkbp:representation_theory |
all irreducible representations are 1-dimensional
|
| gptkbp:simply_connected |
no
|
| gptkbp:structure |
isomorphic to the quotient group R/Z
|
| gptkbp:universalCover |
real line
|
| gptkbp:used_in |
defining characters in harmonic analysis
defining phase in quantum mechanics defining rotations in 2D |
| gptkbp:bfsParent |
gptkb:topology
gptkb:Lie_group gptkb:Lie_groups gptkb:Hopf_fibration gptkb:Unitary_group |
| gptkbp:bfsLayer |
5
|