Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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.

The centralizer of a Cartan element from its vanishing roots

Statement

Let g be a complex semisimple Lie algebra and let g=h⊕⨁α∈Φgα be its root-space decomposition relative to a Cartan subalgebra h. For h∈h,

Cg(h)=h⊕⨁α(h)=0gα.

Facts & Assumptions

Given: A complex semisimple Lie algebra g, an element h∈h in a Cartan subalgebra, and the root-space decomposition of g relative to h.

Proof

technique · direct
1.1

Write x=x0+∑αxα according to Root-space decomposition relative to a Cartan subalgebra, with x0∈h and xα∈gα. Then [h,x]=∑αα(h)xα.

givenalgebra
2.1

The directness of the root-space decomposition implies that [h,x]=0 holds exactly when xα=0 for every root with α(h)≠0. Thus the centralizer consists precisely of h together with the root spaces on which h vanishes.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

2 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