Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Semisimple Lie algebras
Definition
A finite-dimensional Lie algebra is semisimple if its solvable radical vanishes:
Here the radical is the largest solvable ideal from Solvable radical. Under this convention the zero Lie algebra is semisimple, because its radical is zero. A nonzero solvable Lie algebra is not semisimple, since it equals its radical. No assertion about decomposition into simple ideals is built into this definition; that characterization requires later structure theory. The definition is valid over any field and uses no choice principle.
Depends on
Used by
- Centerless does not imply semisimple Counterexample
- Reductive Lie algebras Definition
- Simple, semisimple, and reductive Lie algebras Definition
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Milne, Lie Algebras, Definition 4.2 (standard reference, not scraped)