LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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A nonzero homomorphism between Verma modules is injective

Statement

Every nonzero $g$ -homomorphism $M (μ) \to M (λ)$ is injective.

Facts & Assumptions

Given: The PBW model The PBW model of a Verma module and the domain property The enveloping algebra of the negative nilpotent Lie algebra is a domain.

Proof

technique · direct

1.1

In the PBW identifications, the image of the highest vector is $a v_{λ}$ for a nonzero $a \in U (n^{-})$ , and equivariance makes the map $u v_{μ} \mapsto u a v_{λ}$ .

given

2.1

If $u v_{μ}$ is in the kernel, then $u a = 0$ in $U (n^{-})$ ; the domain property gives $u = 0$ . Hence the kernel is zero.

step 1.1contradiction∎

Depends on

Used by

Dependency tree · two levels

5 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 8.14(i) (standard reference, not scraped)