Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
How 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.

Direct-sum decomposition of a semisimple Lie algebra

Example

Over a characteristic-zero field let g=sl2⊕sl3. Its summands are simple ideals, every ideal is a sum of a subcollection of them, and the Killing form is their orthogonal direct sum.

Facts & Assumptions

Given: The componentwise bracket on the displayed direct sum.

[L1]

Relative to a decomposition of a finite-dimensional semisimple characteristic-zero Lie algebra into simple ideals, every ideal is the sum of a subfamily of the simple factors (Ideals and quotients of semisimple Lie algebras).

Verification

technique · direct
1.1

The standard matrix-unit commutator argument shows that sl2 and sl3 are nonabelian simple in characteristic zero. Hence sl2⊕0 and 0⊕sl3 are simple ideals whose direct sum is g.

givenalgebra
2.1

Applying [L1], the complete ideal list is 0,sl2⊕0,0⊕sl3,g. This includes the empty and full subcollections.

L1step 1.1
3.1

For x∈sl2 and y∈sl3, the two adjoint maps ad⁡(x,0) and ad⁡(0,y) act on opposite blocks, so their product is zero. Restriction to a block is its own adjoint trace. Thus Kg=Ksl2⊥Ksl3. Everything is finite and choice-free.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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