论文标题
virasoro张量类别的结构中央电荷$ 13-6p-6p^{ - 1} $ for Integers $ p> 1 $
Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$
论文作者
论文摘要
令$ \ mathcal {o} _c $为Virasoro Lie代数的有限长度中央电荷 - $ c $模块的类别，其组成因子是可还原的Verma模块的不可减至的商。最近，已经证明$ \ nathcal {o} _c $允许任何$ c \ in \ mathbb {c} $中的任何$ c \ oftertex代数张量类别结构。在这里，当整数$ p> 1 $的$ c = 13-6p-6p^{ - 1} $时，我们确定此张量类别的结构。对于这样的$ c $，我们证明$ \ MATHCAL {O} _ {C} $是刚性的，并且我们在天然张量子子类别$ \ Mathcal {o} _ {c} _ {c}^0 $中构造了不可值模块的投影封面。然后，我们计算所有涉及不可还原模块及其投影覆盖物的张量产品。 Using these tensor product formulas, we show that $\mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$.接下来，作为$ \ Mathcal {o} _C $的编织张量类别结构的直接结果，以及顶点操作员代数扩展理论，我们为三胞胎vertex操作员代数$ \ Mathcal $ \ Mathcal {w}（w}（p）$，包括刚性，固定规则，构造规则和构造覆盖物的封面和构造的封面。最后，我们证明了Negron的最新猜想是$ \ Mathcal {o} _c^0 $编织张量等效于$ psl（2，\ mathbb {c}）$ - equivariant- equivariant of equivariant of equivariant $ \ mathcal {w}（w}（p）$ - 模块的类别。
Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $\mathcal{O}_c$ admits vertex algebraic tensor category structure for any $c\in\mathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $\mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $\mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $\mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $\mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,\mathbb{C})$-equivariantization of the category of $\mathcal{W}(p)$-modules.
