论文标题
联想代数和分级限制顶点代数的表示理论
Associative algebras and the representation theory of grading-restricted vertex algebras
论文作者
论文摘要
我们使用无限矩阵介绍了一个关联代数$ a^{\ infty}（v）$，并在限制级的顶点代数$ v $中输入条目，以使相关的分级空间$ gr（w）= \ coprod_ = \ coprod_ {n \ in \ natbb in \ nathbb {n}} gr _ fill _ fill _ fill_ n} $ V $ -MODULE $ W $是$ a^{\ infty}（v）$ - 模块满足附加属性（称为$ a^{\ infty}（v）$ - 模块）。我们证明，仅当分级$ a^{\ infty}（v）$ - $ - 模块$ gr（w）$分别是不可约或完全还原时，且仅当分级$ a^{\ infty}（v）$ a^{\ iftty}（v）时，较低的广义$ v $ -module $ w $是不可约或完全还原的。我们还证明，较低的广义$ v $模块的一组等价类别与分级$ a^{\ infty}（v）$模块的等价类别的集合进行了培养。对于$ n \ in \ mathbb {n} $，有一个subalgebra $ a^{n}（n}（n}（v）$ a^{\ infty}（v）$，因此子空间$ gr^{n}（n}（w）（w）= \ coprod_ $ a^{n}（v）$ - 模块满足其他属性（称为$ a^{n}（v）$ - 模块）。我们证明$ a^{n}（v）$是有限维数时，当$ v $是正能量（CFT类型）和$ c_ {2} $ - cofinite。我们证明，低结合的$ v $ - 模块的等价类的集合与分级$ a^{n}（n}（v）$模块的等价类的集合进行了两次射击。 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分级$ a^{n}（v）$ - 模块$ gr^{n}（w）$分别是不可约或完全还原的。
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 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 $Gr^{N}(W)$ is irreducible or completely reducible, respectively.
