DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Highest-weight vectors and cyclic highest-weight modules

Definition

Fix a triangular decomposition $g = n^{-} \oplus h \oplus n^{+}$ from Triangular decomposition from a chosen positive root system. Let $M$ be a $g$ -module and let $λ \in h^{*}$ . A nonzero vector $v \in M$ is a highest-weight vector of weight $λ$ when

$h v = λ (h) v (h \in h), n^{+} v = 0.$

A $g$ -module $M$ is a cyclic highest-weight module of highest weight $λ$ when it is generated by such a vector $v$ .

Depends on

Triangular decomposition from a chosen positive root system

Used by

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 (standard reference, not scraped)