sl(2, C) viewed over R real form
Noncompact real form of sl(2, C) ⊕ sl(2, C) with its curated integrating groups.
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 sourcesComplexification
This real form is noncompact. The integration links below identify particular global groups with this local algebra.
Reference properties
One algebra, several integrations
A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel. [sources]
Here the complex algebra is A₁ ⊕ A₁. The same complexification also has the compact real form su(2) ⊕ su(2), but the two real algebras are different: the Lorentz algebra is simple over R and the compact one is a direct sum. [sources]
Related objects and maps
- Actual real Lie algebra
- Complexification
- Curated global form
- Curated global form
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).
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105, printed page 362: complex simple Lie algebras viewed over ROpen source
real-form:sl-2-c-as-real#definitionReal dimension
6
- 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
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
real-form:sl-2-c-as-real#dimensionComplexification
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
real-form:sl-2-c-as-real#complexificationCompact real form
No
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:sl-2-c-as-real#compactReal rank
1
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:sl-2-c-as-real#real-rankOne algebra, several integrations
A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.
- 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
real-form:sl-2-c-as-real#global-forms-1One algebra, several integrations
Here the complex algebra is A₁ ⊕ A₁. The same complexification also has the compact real form su(2) ⊕ su(2), but the two real algebras are different: the Lorentz algebra is simple over R and the compact one is a direct sum.
- 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: real simple classificationOpen source
Premises: algebra:sl-2-c-as-real · algebra:so-4-compact-r · algebra:a-1-plus-a-1-c
real-form:sl-2-c-as-real#global-forms-2Actual real Lie algebra
algebra: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
real-form:sl-2-c-as-real#relation-1Complexification
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
real-form:sl-2-c-as-real#relation-2Curated global form
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
real-form:sl-2-c-as-real#relation-3Curated global form
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
real-form:sl-2-c-as-real#relation-4