Associative algebras and the representation theory of grading-restricted vertex algebras

TL;DR

This paper constructs associative algebras from grading-restricted vertex algebras to classify and analyze their modules, establishing bijections between module categories and proving properties for specific algebra types.

Contribution

It introduces the algebra $A^{ abla}(V)$ and its subalgebras, linking module reducibility to algebra module properties, and proves finite dimensionality under certain conditions.

Findings

Irreducibility of modules corresponds to algebra module irreducibility.

Bijection between module classes and algebra module classes.

Finite dimensionality of $A^{N}(V)$ for $V$ of positive energy and $C_2$-cofinite.

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 $Gr(W)=\coprod_{n\in \mathbb{N}}Gr_{n}(W)$ 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 $N\in \mathbb{N}$, 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 $N\in \mathbb{N}$, 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.