CounterexampleConstruction: 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.

Stability under one generator is not enough

Statement refuted

A subspace stable under one Lie-algebra element is automatically a subrepresentation.

Facts & Assumptions

Given: The standard two-dimensional ${s l}_{2}$ -module over a characteristic-zero field, with basis $v_{+}, v_{-}$ satisfying $e v_{+} = 0$ and $f v_{+} = v_{-}$ .

[L1]

A subrepresentation must be stable under every element of the Lie algebra (Subrepresentations, quotient representations, and intertwiners).

Counterexample

technique · direct

1.1

The line $k v_{+}$ is stable under $e$ , since $e v_{+} = 0 \in k v_{+}$ .

givenalgebra

2.1

It is not stable under $f$ , because $f v_{+} = v_{-}$ and $v_{-}$ is linearly independent from $v_{+}$ . By [L1], $k v_{+}$ is therefore not a subrepresentation.

step 1.1L1algebra

3.1

This line is stable under one named generator but not under the whole Lie algebra, refuting the statement.

step 2.1∎

Depends on

Subrepresentations, quotient representations, and intertwiners

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

Etingof, MIT 18.745 notes, standard sl_2 representation in §11.4, printed pp. 65–69 (standard reference, not scraped)