论文标题
三胞胎顶点操作员代数代数的Yoneda代数代数
Yoneda algebras of the triplet vertex operator algebra
论文作者
论文摘要
给定一个顶点操作员代数$ v $，一个可以构建两个联想代数，Zhu代数$ a（v）$和$ C_2 $ -Algebra $ r（v）$。这引起了两个Abelian类别$ a（v） - \ text {mod} $和$ r（v） - \ text {mod} $，除了$ v $的可允许模块的类别外。如果$ v $是理性的，$ c_2 $ -cofInite，则可接受的$ v $ - 模块类别，所有$ a（v）$ - 模块的类别相当于。但是，当$ v $不是理性的时，这两个类别之间的连接尚不清楚。本文的目的是研究三重态顶点操作员代数$ \ mathcal {w}（p）$，作为比较这三个类别的示例。对于这三个Abelian类别中的每一个，我们将确定相关的ext Quiver，Morita等效的基本代数，即，代数$ \ text {end}（\ oplus_ {l \ in \ in \ text {inr}}} p_l）p_l） $ \ text {ext}^{*}（\ oplus_ {l \ in \ text {irr}} l，\ oplus_ {l \ in \ text {irr}} l）$。结果，三胞胎VOA $ \ MATHCAL W（P）$的可允许的日志模型类别具有无限的全球维度，Zhu代数$ a（\ Mathcal W（p）$也是如此w（p））$。我们还描述了$ \ Mathcal W（p）$，$ a（\ MATHCAL W（p））$和$ \ text {gr} \ a（\ Mathcal W（p））$的模块类别的Koszul属性。
Given a vertex operator algebra $V$, one can construct two associative algebras, the Zhu algebra $A(V)$ and the $C_2$-algebra $R(V)$. This gives rise to two abelian categories $A(V)-\text{Mod}$ and $R(V)-\text{Mod}$, in addition to the category of admissible modules of $V$. In case $V$ is rational and $C_2$-cofinite, the category of admissible $V$-modules and the category of all $A(V)$-modules are equivalent. However, when $V$ is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra $\mathcal{W}(p)$, as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra $ \text{End} (\oplus_{L\in \text{Irr}} P_L)^{op}$, and the Yoneda algebra $\text{Ext}^{*}(\oplus_{L\in \text{Irr}}L, \oplus_{L\in \text{Irr}}L)$. As a consequence, the category of admissible log-modules for the triplet VOA $ \mathcal W(p)$ has infinite global dimension, as do the Zhu algebra $A(\mathcal W(p))$, and the associated graded algebra $\text{gr} \ A(\mathcal W(p))$ which is isomorphic to $R(\mathcal W(p))$. We also describe the Koszul properties of the module categories of $ \mathcal W(p)$, $A(\mathcal W(p))$ and $\text{gr} \ A(\mathcal W(p))$.
