Principal spin-one homomorphism into SU(3)
A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3). Kernel: {±I} ≅ C₂.
A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | su-2 | Sources |
| Codomain | su-3 | Sources |
| Specified homomorphism | A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3). | Sources |
| Kernel | Sources | |
| Image | An SU(3)-conjugate of the real SO(3) subgroup | Sources |
| Injective | No | Sources |
| Surjective | No | Sources |
Global and representation consequences
The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective. [sources]
The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel. [sources]
Related objects and maps
- Domain group
- Codomain group
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
A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#definitionDomain
su-2
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#domainCodomain
su-3
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#codomainSpecified homomorphism
A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#formulaKernel
{±I} ≅ C₂
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#kernelImage
An SU(3)-conjugate of the real SO(3) subgroup
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#imageInjective
No
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#injectiveSurjective
No
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#surjectiveGlobal and representation consequences
The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective.
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen 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:su-2-to-su-3-principal#meaning-1Global and representation consequences
The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel.
- Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
Premises: The defining SU(3) representation restricts to spin 1 · Spin 1 ⊗ spin 1 = spin 0 ⊕ spin 1 ⊕ spin 2
map:su-2-to-su-3-principal#meaning-2Domain group
group:su-2
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#relation-1Codomain group
group:su-3
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) imageOpen source
map:su-2-to-su-3-principal#relation-2