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.

我们证明如果V是一个顶点算子代数，使得所有不可约的普通V-模块都是\（C_1 \）- cofinite，并且V的所有受等级限制的广义Verma模块的长度都是有限的，那么有限长度的范畴就是广义的V模块具有编织张量类别结构。通过将一般性定理应用于与有限简单Lie代数（res。Lie超级代数）\（\ mathfrak {g} \）相关联的简单仿射顶点算子代数（resp。superalgebra）\（\ mathfrak {g} \）在级别k和类别（（KL_k（\ Mathfrak {g}）\）的有限长度广义模块，我们发现了\（KL_k（\ mathfrak {g}）\）在不允许的水平k上，具有编织的张量类别结构。特别地，如果普通模块的类别是半简单的，则\（KL_k（\ mathfrak {g}）\）具有编织张量类别结构，或者如果普通模块的类别是有限的长度，则\（KL_k（\ mathfrak {g}）\）具有编织的张量类别结构。我们还证明了刚度并确定了某些类别\（KL_k（\ mathfrak {g}）\）的融合规则，包括类别\（KL _ {-1}（\ mathfrak {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刚性张量类别结构，该子类别包含具有半简单Cartan子代数作用的对象。