Circle group
The compact connected abelian Lie group of phases. It is the basic model for charge quantization after choosing a period.
Reference properties
Representations
Continuous irreducible complex representations are eⁱθ ↦ eⁱⁿθ with n ∈ ℤ, for θ of period 2π. [sources]
Topology and global form
U(1) is a circle S¹. Its universal covering group is the additive group ℝ. [sources]
In physics
Electromagnetism, phase symmetries, maximal tori, and charge conventions. [sources]
Names and aliases
Circle · SO(2)
Related objects and maps
- Related group
- Related group
- Actual real Lie algebra
- Explicit homomorphism
- Explicit homomorphism
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
Complex numbers z with |z| = 1
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#definitionReal dimension
1
- Lie Groups and Lie AlgebrasSection 6: U(1) has Lie algebra iR of real dimension 1Open source
group:u-1#dimensionCenter
U(1)
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#centerFundamental group
ℤ
- Lie Groups and Lie AlgebrasSection 3.2: R → R/Z as a covering of the circleOpen source
group:u-1#pi1Compact
Yes
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#compactConnected
Yes
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#connectedRank
1
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#rankRepresentations
Continuous irreducible complex representations are eⁱθ ↦ eⁱⁿθ with n ∈ ℤ, for θ of period 2π.
- Conrad and Landesman · Compact Lie groupsLecture 27: characters of a torus; U(1) character lattice ZOpen source
group:u-1#representations-1Topology and global form
U(1) is a circle S¹. Its universal covering group is the additive group ℝ.
- Lie Groups and Lie AlgebrasSection 3.2: covering groups of the circleOpen source
group:u-1#topology-1In physics
Electromagnetism, phase symmetries, maximal tori, and charge conventions.
- Conrad and Landesman · Compact Lie groupsLecture 27: characters of a torus; U(1) character lattice ZOpen source
group:u-1#physics-1Related group
group:su-2
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#relation-1Related group
group:so-3
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-1#relation-2Actual real Lie algebra
algebra:u-1-r
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:u-1#relation-3Explicit homomorphism
map:u-1-to-su-2
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
group:u-1#relation-4Explicit homomorphism
map:u-1-to-so-3
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
group:u-1#relation-5