arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2009.00262math.QAhep-thmath.RAmath.RT
收藏

结合代数与分级受限顶点代数的表示论

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

Yi-Zhi Huang

备注：
44 pages. A mistake in Theorem 4.2 is corrected
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文通过构造与分级受限顶点代数关联的结合代数及其有限维子代数，建立了下界广义顶点代数模与相应分级结合代数模之间的等价对应，并将不可约性与完全可约性转化为结合代数模的相应性质。

摘要：

我们利用以分级受限顶点代数 $V$ 中元素为项的无穷矩阵引入一个结合代数 $A^{\infty}(V)$，使得下界广义 $V$-模 $W$ 的一个滤过的关联分级空间 $Gr(W)=\coprod_{n\in \mathbb{N}}Gr_{n}(W)$ 是一个满足附加性质的 $A^{\infty}(V)$-模（称为分级 $A^{\infty}(V)$-模）。我们证明，下界广义 $V$-模 $W$ 不可约或完全可约当且仅当分级 $A^{\infty}(V)$-模 $Gr(W)$ 分别不可约或完全可约。我们还证明，下界广义 $V$-模的等价类集合与分级 $A^{\infty}(V)$-模的等价类集合之间存在双射。对 $N\in \mathbb{N}$，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（称为分级 $A^{N}(V)$-模）。我们证明，当 $V$ 具有正能量（CFT 型）且 $C_{2}$-余有限时，$A^{N}(V)$ 是有限维的。我们证明，下界广义 $V$-模的等价类集合与分级 $A^{N}(V)$-模的等价类集合之间存在双射。在 $V$ 是 Möbius 顶点代数且不可约下界广义 $V$-模的最低权的实部之差小于或等于 $N\in \mathbb{N}$ 的情形下，我们证明，有限长度的下界广义 $V$-模 $W$ 不可约或完全可约当且仅当分级 $A^{N}(V)$-模 $Gr^{N}(W)$ 分别不可约或完全可约。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑