LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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A Verma embedding implies strong linkage

Statement

Assume the Axiom of Choice.

An embedding $M (μ) ↪ M (λ)$ implies $μ ↑ λ$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), injectivity A nonzero homomorphism between Verma modules is injective, finite length of category $O$ objects Every object of O has finite length, the unique simple Verma quotient A Verma module has a unique simple quotient, and Choice-qualified strong linkage The strong linkage principle for Verma modules.

Proof

technique · direct

1.1given

The image of the embedding is a submodule isomorphic to $M (μ)$ , so its simple quotient $L (μ)$ is a subquotient of $M (λ)$ . Under the stated Choice premise $M (λ)$ has finite length; refining a composition series through this submodule and its maximal proper submodule shows that this simple subquotient is a composition factor.

2.1step 1.1∎

Applying the strong linkage principle to that factor gives $μ ↑ λ$ .

Depends on

Used by

The BGG criterion for homomorphisms between Verma modules Theorem

Dependency tree · two levels

18 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, Corollary 20.14 (standard reference, not scraped)