Normalized maximal torus of SU(2)
eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π. Kernel: {1}.
The group map
Reference properties
Global and representation consequences
The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S². [sources]
The induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator. [sources]
Related objects and maps
- Domain group
- Codomain group
- Associated homogeneous quotient
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
eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.
- 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
map:u-1-to-su-2#definitionDomain
u-1
- 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
map:u-1-to-su-2#domainCodomain
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
map:u-1-to-su-2#codomainSpecified homomorphism
eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.
- 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
map:u-1-to-su-2#formulaKernel
{1}
- 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
map:u-1-to-su-2#kernelImage
{diag(z,z⁻¹) : |z|=1}
- 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
map:u-1-to-su-2#imageInjective
Yes
- 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
map:u-1-to-su-2#injectiveSurjective
No
- 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
map:u-1-to-su-2#surjectiveGlobal and representation consequences
The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S².
- 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
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source
map:u-1-to-su-2#meaning-1Global and representation consequences
The induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator.
- 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
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
Premises: SU(2) is S³ · π₁(U(1))=Z · Fiber loop θ:0→2π · Choice of positively compatible sphere generator
map:u-1-to-su-2#meaning-2Domain group
group:u-1
- 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
map:u-1-to-su-2#relation-1Codomain group
group: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
map:u-1-to-su-2#relation-2Associated homogeneous quotient
space:su-2-over-u-1
- 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
map:u-1-to-su-2#relation-3