U(1)
E553299
U(1) is the group of complex numbers with absolute value 1 under multiplication, commonly representing the symmetry group of electromagnetism and other abelian gauge theories.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| U(1)_Y gauge group | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
abelian group
ⓘ
mathematical group ⓘ one-parameter Lie group ⓘ topological group ⓘ unitary group ⓘ |
| appearsIn |
abelian gauge theories
ⓘ
electromagnetism ⓘ quantum electrodynamics ⓘ |
| hasAlternativeName | circle group ⓘ |
| hasAlternativeNotation | T ⓘ |
| hasCenter | itself ⓘ |
| hasCharacterGroup | Z ⓘ |
| hasChargeQuantizationLabeledBy | integers in many models ⓘ |
| hasCoveringSpace | R with projection \theta \mapsto e^{i\theta} ⓘ |
| hasDefinition | group of complex numbers with absolute value 1 under multiplication ⓘ |
| hasDimension | 1 ⓘ |
| hasElementForm | e^{i\theta} with real \theta ⓘ |
| hasFundamentalGroup | Z ⓘ |
| hasHaarMeasure | normalized Lebesgue measure on the circle ⓘ |
| hasIdentityElement | 1 ⓘ |
| hasInverseOperation | complex conjugation ⓘ |
| hasIrreducibleRepresentationsLabeledBy | integers ⓘ |
| hasLieAlgebra | iR ⓘ |
| hasLieAlgebraDimension | 1 ⓘ |
| hasManifoldStructure | 1-dimensional smooth manifold ⓘ |
| hasOperation | complex multiplication ⓘ |
| hasPontryaginDual | Z ⓘ |
| hasRepresentationTheory | all continuous irreducible unitary representations are 1-dimensional ⓘ |
| hasTopology | subspace topology from C ⓘ |
| isAbelian | true ⓘ |
| isCommutativeLieGroup | true ⓘ |
| isCompact | true ⓘ |
| isConnected | true ⓘ |
| isGaugeGroupOf |
QED
NERFINISHED
ⓘ
electromagnetic interaction ⓘ |
| isHausdorff | true ⓘ |
| isHomeomorphicTo | S^1 ⓘ |
| isIsomorphicTo |
R/Z
ⓘ
SO(2) NERFINISHED ⓘ the circle group NERFINISHED ⓘ |
| isLocallyCompact | true ⓘ |
| isMaximalTorusIn |
SU(2)
ⓘ
U(n) ⓘ |
| isPathConnected | true ⓘ |
| isSimpleConnected | false ⓘ |
| isUnitaryGroupOf | 1-dimensional complex Hilbert space ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
U(1)_Y gauge group