Primitive-exterior-square spin cover in signature (2,3)
For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign. Kernel: {±I} ≅ C₂.
For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | sp-4-r | Sources |
| Codomain | so-2-3-identity | Sources |
| Specified homomorphism | For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign. | Sources |
| Kernel | Sources | |
| Image | Sources | |
| Injective | No | Sources |
| Surjective | Yes | Sources |
| Covering degree | Sources |
Global and representation consequences
The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal. [sources]
On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂. [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
For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#definitionDomain
sp-4-r
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#domainCodomain
so-2-3-identity
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#codomainSpecified homomorphism
For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#formulaKernel
{±I} ≅ C₂
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#kernelImage
SO₀(2,3)
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#imageInjective
No
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#injectiveSurjective
Yes
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#surjectiveCovering degree
2
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#covering-degreeGlobal and representation consequences
The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal.
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen 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:sp-4-r-to-so-2-3#meaning-1Global and representation consequences
On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen 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
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
Premises: U(2)=(SU(2)×U(1))/diagonal C₂ · SU(2)→SO(3) has kernel {±I} · The determinant map induces π₁(U(2))≅Z
map:sp-4-r-to-so-2-3#meaning-2Domain group
group:sp-4-r
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#relation-1Codomain group
group:so-2-3-identity
- Sporadic isogenies to orthogonal groupsSection 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)Open source
- Lie Groups and Lie AlgebrasSection 31.1, printed page 165: the primitive exterior-square symplectic moduleOpen source
map:sp-4-r-to-so-2-3#relation-2