Quaternionic construction of the Spin(5) cover
A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign. Kernel: {±I} ≅ C₂.
A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | sp-2 | Sources |
| Codomain | so-5 | Sources |
| Specified homomorphism | A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign. | Sources |
| Kernel | Sources | |
| Image | Sources | |
| Injective | No | Sources |
| Surjective | Yes | Sources |
| Covering degree | Sources |
Global and representation consequences
The quaternionic compact convention is essential: Sp(2) here is USp(4), not Sp(4,R). The map exhibits Sp(2) ≅ Spin(5). [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 acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#definitionDomain
sp-2
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#domainCodomain
so-5
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#codomainSpecified homomorphism
A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#formulaKernel
{±I} ≅ C₂
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#kernelImage
SO(5)
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#imageInjective
No
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#injectiveSurjective
Yes
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#surjectiveCovering degree
2
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#covering-degreeGlobal and representation consequences
The quaternionic compact convention is essential: Sp(2) here is USp(4), not Sp(4,R). The map exhibits Sp(2) ≅ Spin(5).
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen 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-2-to-so-5#meaning-1Domain group
group:sp-2
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#relation-1Codomain group
group:so-5
- Sporadic isogenies to orthogonal groupsSection 2.6: compact symplectic group acting on traceless quaternionic Hermitian matricesOpen source
map:sp-2-to-so-5#relation-2