LemmaStatement: Literature-sourcedProof: 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 sum of all proper Verma submodules is proper

Statement

The sum $J (λ)$ of all proper submodules of $M (λ)$ is proper.

Facts & Assumptions

Given: The top-weight statement Weights of a Verma module lie below lambda and the highest-line lemma A proper Verma submodule misses the highest-weight line.

Proof

technique · direct

1.1

A submodule is stable under $h$ ; projecting any finite weight decomposition by polynomials in elements of $h$ shows it is the direct sum of its weight intersections.

givenalgebra

2.1

Every proper submodule has zero $λ$ -weight intersection by the highest-line lemma. Hence their sum has zero $λ$ -weight intersection, whereas $v_{λ}$ has that weight; therefore $J (λ) \neq M (λ)$ .

givenalgebra∎

Depends on

Used by

A Verma module has a unique simple quotient Theorem

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

Pavel Etingof, Lie Groups and Lie Algebras I & II, Proposition 25.12 (standard reference, not scraped)