ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 finite-dimensional sl2 quotient of a Verma module

Example

For $m \in Z_{\geq 0}$ ,

$L (m) = M (m) / U ({s l}_{2}) f^{m + 1} v_{m}$

has basis ${\overline{v}}_{m}, \overline{f v_{m}}, \dots, \overline{f^{m} v_{m}}$ and dimension $m + 1$ .

Facts & Assumptions

Given: The unique quotient A Verma module has a unique simple quotient, the action basis The sl2 Verma action in the PBW basis, and the singular vector Reducible and generic sl2 Verma modules.

Verification

technique · direct

1.1

The action formula gives $e f^{m + 1} v_{m} = 0$ , and $f$ and $h$ preserve the span of $f^{n} v_{m}$ for $n \geq m + 1$ ; this is therefore the submodule generated by $f^{m + 1} v_{m}$ . Its quotient has the displayed $m + 1$ surviving, distinct-weight PBW vectors.

givenalgebra

2.1

Any nonzero submodule of the quotient contains a weight vector $\overline{f^{r} v_{m}}$ ; applying $e^{r}$ reaches ${\overline{v}}_{m}$ because all coefficients $k (m - k + 1)$ for $1 \leq k \leq r \leq m$ are nonzero. Applying $f$ then gives every displayed basis vector, so the quotient is simple. The unique-simple-quotient theorem therefore identifies its kernel with $J (m)$ and the quotient with $L (m)$ .

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Pavel Etingof, Representations of Lie Groups, Exercise 8.11 (standard reference, not scraped)