The tensor category for W(2,2)-vertex algebra

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 ${\mathfrak{s}\mathfrak{l}}_2$ Clebsch–Gordan rule. In particular, we prove $$L[r]⊠L[s]\cong \underset{i=0}{\overset{\min \{r,s\}-1}{⨁}}L[r+s-1-2i]\mathrm{.}$$ 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 ${\mathfrak{s}\mathfrak{l}}_2$ of finite dimensional ${\mathfrak{s}\mathfrak{l}}_2$-modules.