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

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Grading-restricted generalized module

Definition

Let $V$ be a conformal vertex algebra of CFT type and $M$ a left $V$ -module. Write $L_{M} (0)$ for the coefficient determined by

Y_{M} (ω, z) = \sum_{n \in Z} L_{M} (n) z^{- n - 2} .

For $h \in C$ , define the generalized $L_{M} (0)$ -eigenspace

M_{[h]} = {m \in M : (L_{M} (0) - h)^{N} m = 0 for some integer N \geq 1} .

The module is grading-restricted generalized if $M = ⨁_{h \in C} M_{[h]}$ , every $M_{[h]}$ is finite dimensional, and the real parts of its nonzero generalized weights are bounded below. The direct sum means each vector has only finitely many generalized-weight components. This convention explicitly adds finite-dimensionality and a lower bound to the generalized-eigenspace decomposition; those conditions are not consequences of the module Jacobi identity.

Depends on

Used by

Logarithmic generalized module Definition
Ordinary module Definition
The Heisenberg Jordan Fock family is grading-restricted; it is logarithmic exactly when λ is nonzero Theorem

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

Yi-Zhi Huang, James Lepowsky, and Lin Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I, Definition 2.12 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc Spaces and Vertex Algebras, conformal-module discussion (standard reference, not scraped)