sl(2, C) ⊕ C — complex Lie algebra
The reductive complex Lie algebra gl(2,C), decomposed into its traceless ideal and scalar center.
The reductive complex Lie algebra gl(2,C), decomposed into its traceless ideal and scalar center.
Definition sourcesReference properties
Using the algebra record
The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information. [sources]
Direct sums and central summands are retained explicitly so semisimple and reductive examples do not appear in the simple-family classification. [sources]
Names and aliases
gl(2,C)
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 reductive complex Lie algebra gl(2,C), decomposed into its traceless ideal and scalar center.
- 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
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
algebra:a-1-plus-abelian-1-c#definitionScalar field
C
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
algebra:a-1-plus-abelian-1-c#fieldComplex dimension
4
- 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
algebra:a-1-plus-abelian-1-c#dimensionComplex rank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
algebra:a-1-plus-abelian-1-c#rankRoot-system type
A₁ plus center
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
algebra:a-1-plus-abelian-1-c#cartan-typeSimple over C
No
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
algebra:a-1-plus-abelian-1-c#simpleUsing the algebra record
The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.
- 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:a-1-plus-abelian-1-c#interpretation-1Using the algebra record
Direct sums and central summands are retained explicitly so semisimple and reductive examples do not appear in the simple-family classification.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
algebra:a-1-plus-abelian-1-c#interpretation-2