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

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 $Gr(W)={\coprod }_{n\in \mathbb{N}}G{r}_n(W)$ 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 $G{r}^N(W)={\coprod }_{n=0}^{N}G{r}_n(W)$ 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 Möbius 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 $G{r}^N(W)$ is irreducible or completely reducible, respectively.

44 pages. A mistake in Theorem 4.2 is corrected

References in corpus (1)

• A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra