论文标题
张量的仿射类别谎言代数超出了可允许的水平
Tensor categories of affine Lie algebras beyond admissible levels
论文作者
论文摘要
我们表明，如果$ v $是一个顶点操作员代数，那么所有不可修复的普通$ v $ - 模块都是$ C_1 $ -COFINITE，所有$ V $的分级限制的广义Verma模块都是有限长度的，那么有限长度的$ V $ -V $ -MODULES类别是一个有限的$ -Modules类别。 通过将一般定理应用于与有限的简单的lie代数相关的简单仿射顶点操作员代数（分别superalgebra）（superalgebra）$ \ mathfrak {g} $ l级$ k $和类别$ kl_k（\ nathfrak {g}）的类别（\ m athfrak {g}）$ $ kl_k（\ Mathfrak {g}）$在不可加速的级别$ k $，具有编织张量类别结构。特别是，如果普通模块的类别是半缩影，则$ kl_k（\ mathfrak {g}）$，如果普通模块的类别具有有限的长度，则具有编织的张量类别结构。 我们还证明了刚度并确定某些类别的融合规则$ kl_k（\ mathfrak {g}）$，包括类别$ kl _ { - 1}（\ Mathfrak {sl sl} _n）$。使用这些结果，我们在$ kl_1（\ mathfrak {sl}（n | m））的完整子类别上构造了一个刚性张量的类别结构，该子类由具有半密码cartan子词法动作的对象组成。
We show that if $V$ is a vertex operator algebra such that all the irreducible ordinary $V$-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for $V$ are of finite length, then the category of finite length generalized $V$-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $\mathfrak{g}$ at level $k$ and the category $KL_k(\mathfrak{g})$ of its finite length generalized modules, we discover several families of $KL_k(\mathfrak{g})$ at non-admissible levels $k$, having braided tensor category structures. In particular, $KL_k(\mathfrak{g})$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $KL_k(\mathfrak{g})$, including the category $KL_{-1}(\mathfrak{sl}_n)$. Using these results, we construct a rigid tensor category structure on a full subcategory of $KL_1(\mathfrak{sl}(n|m))$ consisting of objects with semisimple Cartan subalgebra actions.
