sl(2, C) viewed over R — real Lie algebra
The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).
The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).
Definition sourcesReference properties
From algebra to group
The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree. [sources]
Restriction of scalars doubles the real dimension. Complexifying this realification gives two complex sl(2,C) factors, which label the two chiral highest weights. It is not a real form of the single complex A₁ algebra. [sources]
Names and aliases
so(1,3) · so(3,1)
Related objects and maps
- Complexification
- Real-form context
- Curated integrating group
- Curated integrating 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
The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).
- 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
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105, printed page 362: complex simple Lie algebras viewed as simple real algebrasOpen source
algebra:sl-2-c-as-real#definitionScalar field
R
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
algebra:sl-2-c-as-real#fieldReal dimension
6
- 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
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
algebra:sl-2-c-as-real#dimensionAbsolute rank (of complexification)
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
algebra:sl-2-c-as-real#rankSimple over R
Yes
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over ROpen source
algebra:sl-2-c-as-real#simpleCompact real algebra
No
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:sl-2-c-as-real#compactComplexification
sl(2, C) ⊕ sl(2, C)
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:sl-2-c-as-real#complexificationReal rank
1
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:sl-2-c-as-real#real-rankFrom algebra to group
The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen 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
algebra:sl-2-c-as-real#integration-1From algebra to group
Restriction of scalars doubles the real dimension. Complexifying this realification gives two complex sl(2,C) factors, which label the two chiral highest weights. It is not a real form of the single complex A₁ algebra.
- Lie Groups and Lie AlgebrasSection 9.4: real forms and complexificationOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105: complex simple algebras considered as real algebrasOpen source
Premises: algebra:a-1-c · Restriction of scalars R ⊂ C
algebra:sl-2-c-as-real#integration-2Complexification
algebra:a-1-plus-a-1-c
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:sl-2-c-as-real#relation-1Real-form context
real-form:sl-2-c-as-real
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:sl-2-c-as-real#relation-2Curated integrating group
group:sl-2-c
- 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 groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
algebra:sl-2-c-as-real#relation-3Curated integrating group
group:so-1-3-identity
- 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 groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
algebra:sl-2-c-as-real#relation-4