Regular block inclusion in SU(3)
A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed. Kernel: {I}.
A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.
Definition sourcesThe group map
Reference properties
Global and representation consequences
The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1. [sources]
For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet. [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 ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#definitionDomain
su-2
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#domainCodomain
su-3
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#codomainSpecified homomorphism
A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#formulaKernel
{I}
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#kernelImage
{diag(A,1) : A ∈ SU(2)} ≅ SU(2)
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#imageInjective
Yes
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#injectiveSurjective
No
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#surjectiveGlobal and representation consequences
The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1.
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open 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-regular#meaning-1Global and representation consequences
For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet.
- Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
- Haber · Spin covers and subgroup examplesProblem 5: regular inclusionOpen source
Premises: 3 restricts to 2⊕1 · Adjoint SU(3) = End(3) minus scalars
map:su-2-to-su-3-regular#meaning-2Domain group
group:su-2
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#relation-1Codomain group
group:su-3
- Haber · Spin covers and subgroup examplesProblem 5, pages 14–15: block SU(2) subgroup of SU(3)Open source
map:su-2-to-su-3-regular#relation-2