Vector spin cover in eight Euclidean dimensions
g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8). Kernel: {±1} ≅ C₂.
g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).
Definition sourcesThe group map
Reference properties
Global and representation consequences
This specifies the vector-cover kernel inside the full C₂ × C₂ center. Triality changes the role of vector and half-spin modules and permutes the three order-two kernels. [sources]
Although the three quotients by order-two central subgroups are abstractly isomorphic, a representation descends according to the kernel of the specified map. The quotient name alone does not retain this choice. [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
g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).
- 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
map:spin-8-to-so-8#definitionDomain
spin-8
- 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
map:spin-8-to-so-8#domainCodomain
so-8
- 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
map:spin-8-to-so-8#codomainSpecified homomorphism
g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).
- 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
map:spin-8-to-so-8#formulaKernel
{±1} ≅ C₂
- 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
map:spin-8-to-so-8#kernelImage
SO(8)
- 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
map:spin-8-to-so-8#imageInjective
No
- 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
map:spin-8-to-so-8#injectiveSurjective
Yes
- 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
map:spin-8-to-so-8#surjectiveCovering degree
2
- 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
map:spin-8-to-so-8#covering-degreeGlobal and representation consequences
This specifies the vector-cover kernel inside the full C₂ × C₂ center. Triality changes the role of vector and half-spin modules and permutes the three order-two kernels.
- 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
- 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:spin-8-to-so-8#meaning-1Global and representation consequences
Although the three quotients by order-two central subgroups are abstractly isomorphic, a representation descends according to the kernel of the specified map. The quotient name alone does not retain this choice.
- Conrad and Landesman · Compact Lie groupsRemark T.2.1, printed page 262: D₄ triality and quotient groupsOpen source
map:spin-8-to-so-8#meaning-2Domain group
group:spin-8
- 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
map:spin-8-to-so-8#relation-1Codomain group
group:so-8
- 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
map:spin-8-to-so-8#relation-2