论文标题
大模块类别的刚性张量结构，用于某些$ W $ - （超级）代数$ a $
Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
论文作者
论文摘要
我们在有限生成的重量模块上建立刚性张量的类别结构，用于$ \ mathfrak {sl} _n $的次规格$ w $ - 代理，$ - n+\ n+\ frac {n} {n} {n+1} $ n + \ frac {n + 1} {n} $（$βγ$ -Vertex代数的有限环形），以及其Feigin-Semikhatov Dual dual princmal $ w $ w $ w $ -superalgebras $ \ superalgebras的$ \ mathfrak {sl sl} _ {这些类别既不是有限的，也不是半圆形，在$ W $ - 代数案件中，它们包含具有无限尺寸保形重量空间的模块，并且在保形重量上没有下限。我们在这些张量类别中提供了不可分解的投影模块的完整列表，并为简单模块提供了融合规则。所有这些顶点运算符（超级）代数都是用高级海森贝格代数张紧的单元代数的简单电流扩展，因此我们更普遍地研究了顶点代数张量张量类别的直接限制完成中的简单当前电流扩展。然后，我们的$ W $ - （超级）代数的结果来自Singlet代数的模块上已知的色带类别结构。我们的结果包括并概括了Allen-Wood在$βγ$ -VERTEX代数上的结果，以及我们自己在$ \ mathfrak {gl} _ {1 | 1 | 1} $的offine顶点superalgebra上。我们的结果还包括在所有有限生成的重量模块上，在非综合允许水平上的所有有限生成的重量模块上的第一个示例
We establish rigid tensor category structure on finitely-generated weight modules for the subregular $W$-algebras of $\mathfrak{sl}_n$ at levels $ - n + \frac{n}{n+1}$ (the $\mathcal{B}_{n+1}$-algebras of Creutzig-Ridout-Wood) and at levels $- n + \frac{n+1}{n}$ (the finite cyclic orbifolds of the $βγ$-vertex algebra), as well as for their Feigin-Semikhatov dual principal $W$-superalgebras of $\mathfrak{sl}_{n|1}$. These categories are neither finite nor semisimple, and in the $W$-algebra case they contain modules with infinite-dimensional conformal weight spaces and no lower bound on conformal weights. We give complete lists of indecomposable projective modules in these tensor categories and fusion rules for simple modules. All these vertex operator (super)algebras are simple current extensions of singlet algebras tensored with a rank-one Heisenberg algebra, so we more generally study simple current extensions in direct limit completions of vertex algebraic tensor categories. Then our results for $W$-(super)algebras follow from the known ribbon category structure on modules for the singlet algebras. Our results include and generalize those of Allen-Wood on the $βγ$-vertex algebra, as well as our own on the affine vertex superalgebra of $\mathfrak{gl}_{1|1}$. Our results also include the first examples of ribbon category structure on all finitely-generated weight modules for an affine vertex algebra at a non-integral admissible level, namely for affine $\mathfrak{sl}_2$ at levels $-\frac{4}{3}$ and $-\frac{1}{2}$.
