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.