Lie theory · Group
Compact group of octonion automorphisms
Simply connected and adjoint compact form of G₂. The automorphism group of the real octonion division algebra O.
Executable coverageCompact Lie groupSimple algebraSimply connected
Reference properties
Global structure and topology
The compact G₂ group is simply connected and has trivial center, so its simply connected and adjoint forms coincide. [sources]
Representation category and descent
Octonion automorphisms fix the real unit and preserve the imaginary octonions. Their action on Im(O) is a faithful real representation of dimension 7, preserving its Euclidean norm and octonion multiplication. [sources]
Names and aliases
Aut(O) · Compact G2
Related objects and maps
- Real Lie algebra
- Real-form context
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
The automorphism group of the real octonion division algebra O.
- The OctonionsSection 4.1: G₂ as octonion automorphismsOpen source
group:g-2-compact#definitionReal dimension
14
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
- Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source
group:g-2-compact#dimensionCompact rank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:g-2-compact#rankCenter
1
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
group:g-2-compact#centerFundamental group π₁
0
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen 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
group:g-2-compact#pi1Compact
Yes
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:g-2-compact#compactConnected
Yes
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:g-2-compact#connectedAbelian
No
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:g-2-compact#abelianSimple real Lie algebra
Yes
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:g-2-compact#simple-algebraGlobal form
Simply connected and adjoint compact form of G₂
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:g-2-compact#global-formReal Lie algebra
g₂ compact
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:g-2-compact#lie-algebraGlobal structure and topology
The compact G₂ group is simply connected and has trivial center, so its simply connected and adjoint forms coincide.
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:g-2-compact#topology-1Representation category and descent
Octonion automorphisms fix the real unit and preserve the imaginary octonions. Their action on Im(O) is a faithful real representation of dimension 7, preserving its Euclidean norm and octonion multiplication.
- The OctonionsSection 4.1: action on imaginary octonions and the subgroup G₂ ⊂ SO(7)Open source
group:g-2-compact#representations-1Real Lie algebra
algebra:g-2-compact-r
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:g-2-compact#relation-1Real-form context
real-form:g-2-compact-r
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:g-2-compact#relation-2