DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Admissible module for a vertex operator algebra

Definition

Let $V = ⨁_{r \geq 0} V_{r}$ be a conformal vertex algebra of CFT type and $M$ a left $V$ -module. The module is admissible if it is equipped with a direct-sum grading $M = ⨁_{n \geq 0} M (n)$ such that for every homogeneous $a \in V_{r}$ , every integer mode $k$ , and every $n \geq 0$ ,

a_{(k)}^{M} M (n) \subseteq M (n + r - k - 1),

where $M (q) = 0$ for $q < 0$ . No diagonalizability of $L_{M} (0)$ is required: the conformal mode $L_{M} (0) = ω_{(1)}^{M}$ preserves each $M (n)$ , but may act there nonsemisimply. In this definition, morphisms preserve the specified nonnegative grading and commute with every module field.

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

Chongying Dong and Cuipo Jiang, Rationality of Vertex Operator Algebras, §2 (standard reference, not scraped)
Yi-Zhi Huang, James Lepowsky, and Lin Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I, §2 (standard reference, not scraped)