u(1) — real Lie algebra
The real one-dimensional abelian Lie algebra iR with zero bracket. Its abstract Lie algebra alone does not distinguish the integrating groups U(1) and R.
The real one-dimensional abelian Lie algebra iR with zero bracket. Its abstract Lie algebra alone does not distinguish the integrating groups U(1) and R.
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]
For an abelian central summand, the abstract real Lie algebra does not encode whether the corresponding connected group factor is a circle or an additive real line. This record therefore links the chosen compact integration explicitly. [sources]
Names and aliases
so(2) · R with zero bracket
Related objects and maps
- Complexification
- 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 real one-dimensional abelian Lie algebra iR with zero bracket. Its abstract Lie algebra alone does not distinguish the integrating groups U(1) and 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
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
algebra:u-1-r#definitionScalar field
R
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
algebra:u-1-r#fieldReal dimension
1
- 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:u-1-r#dimensionAbsolute rank (of complexification)
1
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
algebra:u-1-r#rankSimple over R
No
- 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:u-1-r#simpleCompact real algebra
Yes
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:u-1-r#compactComplexification
C (zero bracket)
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:u-1-r#complexificationFrom 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:u-1-r#integration-1From algebra to group
For an abelian central summand, the abstract real Lie algebra does not encode whether the corresponding connected group factor is a circle or an additive real line. This record therefore links the chosen compact integration explicitly.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
algebra:u-1-r#integration-2Complexification
algebra:abelian-1-c
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
algebra:u-1-r#relation-1Curated integrating group
group:u-1
- 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:u-1-r#relation-2