Lie theory · Map
Vector spin cover in ten Euclidean dimensions
g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra. Kernel: {±1} ≅ C₂.
Reference coverageCovering or central quotient
The group map
Reference properties
Global and representation consequences
The full center of Spin(10) is C₄, while this map has the unique order-two kernel. The half-spin 16s do not descend through it. [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 R¹⁰ inside its Euclidean Clifford algebra.
- 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-10-to-so-10#definitionDomain
spin-10
- 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-10-to-so-10#domainCodomain
so-10
- 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-10-to-so-10#codomainSpecified homomorphism
g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra.
- 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-10-to-so-10#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-10-to-so-10#kernelImage
SO(10)
- 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-10-to-so-10#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-10-to-so-10#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-10-to-so-10#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-10-to-so-10#covering-degreeGlobal and representation consequences
The full center of Spin(10) is C₄, while this map has the unique order-two kernel. The half-spin 16s do not descend through it.
- 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-10-to-so-10#meaning-1Domain group
group:spin-10
- 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-10-to-so-10#relation-1Codomain group
group:so-10
- 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-10-to-so-10#relation-2