PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07

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.

Verma homomorphisms and singular vectors

Statement

For weights $λ, μ$ , evaluation at the highest-weight vector gives a natural vector-space isomorphism ${Hom}_{g} (M (μ), M (λ)) ≅ {v \in M (λ)_{μ} : n^{+} v = 0} .$

Facts & Assumptions

Given: The universal property of The universal property of Verma modules.

Proof

technique · direct

1.1given

A homomorphism $f$ sends the highest-weight vector $v_{μ}$ to a vector of weight $μ$ killed by $n^{+}$ ; evaluation is therefore a linear map into the displayed space.

2.1givenconstruct∎

Conversely, a vector $v$ in that space is a highest-weight vector of weight $μ$ , so the universal property supplies a unique homomorphism $M (μ) \to M (λ)$ sending $v_{μ}$ to $v$ . The two constructions are inverse.

Depends on

The universal property of Verma modules

Used by

Peeling a maximal-weight Verma from a standard filtration Lemma

Dependency tree · two levels

3 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, §15.1 (standard reference, not scraped)