论文标题
单元顶点代数的完整表示类别的带张量结构
Ribbon tensor structure on the full representation categories of the singlet vertex algebras
论文作者
论文摘要
我们表明，单点顶点代数$ \ Mathcal {m}（p）$，$ p \ in \ Mathbb {z} _ {> 1} $等于类别$ \ nathcal {o} _ {\ nime， $ \ MATHCAL {M}（P）$ - 模块，并且此类别允许Huang-Lepowsky-Zhang的顶点代数编织的张量类别结构。 Since $\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical $\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical $ \ MATHCAL {M}（P）$ - 模块。我们还引入了一个张量子子类别$ \ MATHCAL {o} _ {\ MATHCAL {M}（P）}^T $，由代数的Torus $ t $分级，该代数具有足够的投影剂，并且具有足够的投射率，并且具有指定性的张量，可与未量化的量化量的量化量的量化类别相等， $ \ mathfrak {sl} _2 $在$ 2p $ th unity的根部。我们计算所有涉及简单和投影的张量产品$ \ MATHCAL {m}（p）$ - 模块，我们证明这两个张量类别$ \ MATHCAL {o} _ {\ MATHCAL {M MATHCAL {M}（M}（M}（P）}（P）} $ and $ \ MATHCAL {O} _} $ and} 丝带。作为应用程序，我们使用顶点操作符代数扩展理论来表明三胞胎顶点代数$ \ mathcal {w}（p）$的所有有限环节的表示类别类别Orbifolds。
We show that the category of finite-length generalized modules for the singlet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to the category $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite $\mathcal{M}(p)$-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since $\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical $\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical $\mathcal{M}(p)$-modules. We also introduce a tensor subcategory $\mathcal{O}_{\mathcal{M}(p)}^T$, graded by an algebraic torus $T$, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of $\mathfrak{sl}_2$ at a $2p$th root of unity. We compute all tensor products involving simple and projective $\mathcal{M}(p)$-modules, and we prove that both tensor categories $\mathcal{O}_{\mathcal{M}(p)}$ and $\mathcal{O}_{\mathcal{M}(p)}^T$ are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.
