Exterior-square spin cover in signature (2,4)
Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign. Kernel: {±I} ≅ C₂.
Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | su-2-2 | Sources |
| Codomain | so-2-4-identity | Sources |
| Specified homomorphism | Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign. | Sources |
| Kernel | Sources | |
| Image | Sources | |
| Injective | No | Sources |
| Surjective | Yes | Sources |
| Covering degree | Sources |
Global and representation consequences
This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group. [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
Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#definitionDomain
su-2-2
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#domainCodomain
so-2-4-identity
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#codomainSpecified homomorphism
Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#formulaKernel
{±I} ≅ C₂
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#kernelImage
SO₀(2,4)
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#imageInjective
No
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#injectiveSurjective
Yes
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#surjectiveCovering degree
2
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#covering-degreeGlobal and representation consequences
This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group.
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)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-2-2-to-so-2-4#meaning-1Domain group
group:su-2-2
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#relation-1Codomain group
group:so-2-4-identity
- Sporadic isogenies to orthogonal groupsSection 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
map:su-2-2-to-so-2-4#relation-2