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- W3082091497 abstract "We introduce an associative algebra $A^{infty}(V)$ using infinite matrices with entries in a grading-restricted vertex algebra $V$ such that the associated graded space $Gr(W)=coprod_{nin mathbb{N}}Gr_{n}(W)$ of a filtration of a lower-bounded generalized $V$-module $W$ is an $A^{infty}(V)$-module satisfying additional properties (called a graded $A^{infty}(V)$-module). We prove that a lower-bounded generalized $V$-module $W$ is irreducible or completely reducible if and only if the graded $A^{infty}(V)$-module $Gr(W)$ is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized $V$-modules are in bijection with the set of the equivalence classes of graded $A^{infty}(V)$-modules. For $Nin mathbb{N}$, there is a subalgebra $A^{N}(V)$ of $A^{infty}(V)$ such that the subspace $Gr^{N}(W)=coprod_{n=0}^{N}Gr_{n}(W)$ of $Gr(W)$ is an $A^{N}(V)$-module satisfying additional properties (called a graded $A^{N}(V)$-module). We prove that $A^{N}(V)$ are finite dimensional when $V$ is of positive energy (CFT type) and $C_{2}$-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized $V$-modules is in bijection with the set of the equivalence classes of graded $A^{N}(V)$-modules. In the case that $V$ is a M{o}bius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized $V$-modules are less than or equal to $Nin mathbb{N}$, we prove that a lower-bounded generalized $V$-module $W$ of finite length is irreducible or completely reducible if and only if the graded $A^{N}(V)$-module $Gr^{N}(W)$ is irreducible or completely reducible, respectively." @default.
- W3082091497 created "2020-09-08" @default.
- W3082091497 creator A5066095681 @default.
- W3082091497 date "2020-09-01" @default.
- W3082091497 modified "2023-10-12" @default.
- W3082091497 title "Associative algebras and the representation theory of grading-restricted vertex algebras" @default.
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- W3082091497 doi "https://doi.org/10.48550/arxiv.2009.00262" @default.
- W3082091497 hasPublicationYear "2020" @default.
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