AI 中文总结
本文研究W(2,2)-顶点算子代数相关的辫子张量范畴𝒞，证明其单对象融合规则符合𝔰𝔩₂的Clebsch-Gordan规则，建立𝒞的刚性，且其半单子范畴与Rep 𝔰𝔩₂张量等价。
AI 中文摘要
本文研究与W-代数W(2,2)相关的顶点算子代数的分级受限C₁余有限广义模的范畴𝒞。我们首先证明𝒞与有限长度模的范畴一致，其单合成因子为最高权( (1-r)/2, 0 )的不可约最高权W(2,2)-模L[r]（r∈正整数），因此𝒞具有辫子张量范畴结构。随后证明单对象L[r]的融合规则由𝔰𝔩₂的Clebsch-Gordan规则支配，特别地，我们证明L[r]⊠L[s]≅⊕_{i=0}^{min{r,s}-1} L[r+s-1-2i]。利用融合规则及Etingof-Penneys的最新结果，我们建立𝒞的刚性，还证明由单对象L[r]生成的半单子范畴与有限维𝔰𝔩₂-模的范畴Rep 𝔰𝔩₂张量等价。
英文摘要
This paper studies the category $\mathcal{C}$ of grading-restricted $C_1$-cofinite generalized modules for the vertex operator algebra associated to the $W$-algebra $W(2,2)$. We first show that $\mathcal{C}$ is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight $W(2,2)$--modules $L[r]$ of highest weight $(\frac{1-r}{2}, 0)$ for $r \in \mathbb{Z}_{> 0}$, and hence $\mathcal{C}$ carries a braided tensor category structure. Then we prove the fusion rules for the simple objects $L[r]$ are governed by the $\mathfrak{sl}_2$ Clebsch--Gordan rule. In particular, we prove \[ L[r]\boxtimes L[s]\cong \bigoplus_{i=0}^{\min\{r,s\}-1} L[r+s-1-2i]. \] Using the fusion rules and a recent result of Etingof--Penneys, we establish the rigidity of $\mathcal{C}$. We also show the semisimple subcategory generated by the simple objects $L[r]$ is tensor equivalent to the category Rep $\mathfrak{sl}_2$ of finite dimensional $\mathfrak{sl}_2$-modules.
