Alphabeta Math
False statementConstruction: AI-adaptedVerification: 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.

An enveloping algebra need not be commutative

Statement

U(g) is commutative for every Lie algebra g.

Facts & Assumptions

Given: The asserted universal commutativity.

[L1]

In U(g), the commutator of canonical images is the image of the Lie bracket (The canonical map to U(g) is a Lie homomorphism).

[L2]

PBW makes the canonical map injective when a basis is supplied (The canonical map g→U(g) is injective).

Refutation

technique · direct counterexample
1.1constructL1

Let g have basis h,e with [h,e]=e, for example the matrix Lie algebra used in the preceding false statement. By [L1], ιg(h)ιg(e)−ιg(e)ιg(h)=ιg(e).

2.1step 1.1L2∎

The supplied basis lets [L2] show ιg(e)≠0. Thus the two elements ιg(h) and ιg(e) do not commute in U(g), refuting the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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