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 sl2 Verma action in the PBW basis

Example

For ${s l}_{2} = ⟨ e, f, h ⟩$ with $[h, f] = - 2 f$ and $[e, f] = h$ , the PBW basis of $M (λ)$ is $f^{n} v_{λ}$ ( $n \geq 0$ ), and

$h f^{n} v_{λ} = (λ - 2 n) f^{n} v_{λ}, f f^{n} v_{λ} = f^{n + 1} v_{λ} (n \geq 0),$

and

$e v_{λ} = 0, e f^{n} v_{λ} = n (λ - n + 1) f^{n - 1} v_{λ} (n \geq 1) .$

Facts & Assumptions

Given: The induced highest vector Verma modules and its PBW model The PBW model of a Verma module.

Verification

technique · direct

1.1

The first two identities follow by $[h, f] = - 2 f$ and multiplication; at $n = 0$ , $e v_{λ} = 0$ .

givenalgebra

2.1

Induction using $e f^{n + 1} = f e f^{n} + h f^{n}$ transforms the $n$ th coefficient into $(n + 1) (λ - n)$ , which is the stated formula for $n + 1$ .

givenalgebra∎

Depends on

Used by

Reducible and generic sl2 Verma modules Example
The finite-dimensional sl2 quotient of a Verma module Example
The sl2 Shapovalov norm product Example

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

Mrudul Thatte, Category O: Verma's Thesis, §1.2 (standard reference, not scraped)