CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

Symmetrization does not preserve products in ${s l}_{2}$

Statement refuted

PBW symmetrization preserves products in ${s l}_{2}$ .

Facts & Assumptions

Given: ${s l}_{2}$ over a characteristic-zero field, with $[h, e] = 2 e$ .

[L1]

Symmetrization satisfies $sym (e h) = \frac{1}{2} (e h + h e)$ and is a vector-space isomorphism (PBW symmetrization in characteristic zero).

Counterexample

technique · direct computation

1.1givenL1algebra

In $U ({s l}_{2})$ , $h e - e h = 2 e$ , so $sym (e h) = \frac{1}{2} (e h + h e) = e h + e$ .

2.1step 1.1L1algebra

On degree-one factors, $sym (e) sym (h) = e h$ . PBW injectivity from [L1] gives $e \neq 0$ in the enveloping algebra, so $e h + e \neq e h$ .

3.1step 2.1∎

Therefore symmetrization does not preserve this product and is not an algebra homomorphism.

Depends on

PBW symmetrization in characteristic zero

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Etingof, MIT 18.745 notes, Corollary 13.7 and Example 13.9, printed p. 75 (standard reference, not scraped)

Symmetrization does not preserve products in sl2sl2​