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

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 Mobius 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.

我们利用在一个分次限制的顶点代数\(V\)中取值的无穷矩阵引入一个结合代数\(A^{\infty}(V)\)，使得一个有下界的广义\(V\)-模\(W\)的一个滤过的相关分次空间\(Gr(W)=\coprod_{n\in \mathbb{N}}Gr_{n}(W)\)是一个满足附加性质的\(A^{\infty}(V)\)-模（称为分次\(A^{\infty}(V)\)-模）。我们证明一个有下界的广义\(V\)-模\(W\)是不可约的或者完全可约的，当且仅当分次\(A^{\infty}(V)\)-模\(Gr(W)\)分别是不可约的或者完全可约的。我们还证明有下界的广义\(V\)-模的等价类集合与分次\(A^{\infty}(V)\)-模的等价类集合是一一对应的。对于\(N\in \mathbb{N}\)，存在\(A^{\infty}(V)\)的一个子代数\(A^{N}(V)\)，使得\(Gr(W)\)的子空间\(Gr^{N}(W)=\coprod_{n = 0}^{N}Gr_{n}(W)\)是一个满足附加性质的\(A^{N}(V)\)-模（称为分次\(A^{N}(V)\)-模）。我们证明当\(V\)是正能量（共形场论类型）且\(C_{2}\)-余有限时，\(A^{N}(V)\)是有限维的。我们证明有下界的广义\(V\)-模的等价类集合与分次\(A^{N}(V)\)-模的等价类集合是一一对应的。在\(V\)是一个莫比乌斯顶点代数且不可约有下界的广义\(V\)-模的最低权的实部之间的差小于或等于\(N\in \mathbb{N}\)的情形下，我们证明一个有限长度的有下界的广义\(V\)-模\(W\)是不可约的或者完全可约的，当且仅当分次\(A^{N}(V)\)-模\(Gr^{N}(W)\)分别是不可约的或者完全可约的。