su(2,2) ≅ so(2,4) real form
Noncompact real form of sl(4, C) with its curated integrating groups.
The real algebra of traceless complex 4 × 4 matrices X satisfying X†H + HX = 0 for H = diag(1,1,−1,−1). It is isomorphic to so(2,4).
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]
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 real algebra of traceless complex 4 × 4 matrices X satisfying X†H + HX = 0 for H = diag(1,1,−1,−1). It is isomorphic to so(2,4).
- 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
real-form:su-2-2-r#definitionReal dimension
15
- 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:su-2-2-r#dimensionComplexification
sl(4, C)
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:su-2-2-r#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:su-2-2-r#compactReal rank
2
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:su-2-2-r#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:su-2-2-r#global-forms-1Actual real Lie algebra
algebra:su-2-2-r
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:su-2-2-r#relation-1Complexification
algebra:a-3-c
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
real-form:su-2-2-r#relation-2Curated global form
group:su-2-2
- 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:su-2-2-r#relation-3Curated global form
group:so-2-4-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:su-2-2-r#relation-4