sl(2, C) ⊕ sl(2, C) — complex Lie algebra
The direct sum of two complex sl(2,C) ideals; equivalently so(4,C). This algebra is semisimple and not simple.
The direct sum of two complex sl(2,C) ideals; equivalently so(4,C). This algebra is semisimple and not simple.
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
so(4,C) · D₂
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 direct sum of two complex sl(2,C) ideals; equivalently so(4,C). This algebra is semisimple and not simple.
- 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-a-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-a-1-c#fieldComplex dimension
6
- 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-a-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-a-1-c#rankRoot-system type
A₁ ⊕ A₁
- 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-a-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-a-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-a-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-a-1-c#interpretation-2