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Quasi-finite-dimensional generalized module

Statement

Let $W = ⨁_{h \in R} W_{[h]}$ be a generalized V-module with real generalized L(0)-weights. It is quasi-finite-dimensional if for every real N the strict cutoff sum $⨁_{h < N} W_{[h]}$ is finite dimensional.

Facts & Assumptions

Given: Work over C with the formal-variable, grading, and branch conventions in the linked definitions.

[F1]

Grading-restricted generalized module: 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.

Proof

1.1givenF1algebra

The generalized-weight decomposition makes each strict cutoff sum a defined subspace of W.

2.1givenF1step 1.1algebra∎

Assume the quasi-finite condition. Each $W_{[h]}$ is contained in the finite-dimensional cutoff sum for $N = h + 1$ , so each weight space is finite dimensional. The cutoff sum for $N = 0$ can contain only finitely many nonzero weight spaces: vectors from distinct summands are linearly independent, so their number is at most the dimension of that cutoff. If there are negative weights, their finite nonempty set has a least element $B$ ; otherwise put $B = 0$ . Every nonzero weight is then at least $B$ . Thus quasi-finite-dimensionality implies both finite-dimensional weight spaces and a global lower bound, and hence grading restriction in the real-weight setting of F1. The definition retains the additional requirement that every cutoff sum be finite dimensional.

Quasi-finite-dimensional generalized module

Statement

Facts & Assumptions

Proof

Depends on

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