Tensor categories of weight modules of sl ̂ 2 $\widehat{\mathfrak {sl}}_2$ at admissible level

The category of weight modules $L_k\left({\mathrm{sl}}_2\right)-wtmod$ of the simple affine vertex algebra of ${\mathrm{sl}}_2$ at an admissible level $k$ is neither finite nor semisimple and modules are usually not lower-bounded and have infinite-dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with $W_{\ell }\left({\mathrm{sl}}_{2|1}\right)$, the simple principal $W$-algebra of ${\mathrm{sl}}_{2|1}$ at level $\ell$ (the $N=2$ super conformal algebra) where the levels are related via $(k+2)(\ell +1)=1$. Every weight module of $W_{\ell }\left({\mathrm{sl}}_{2|1}\right)$ is lower-bounded and has finite-dimensional conformal weight spaces. The main technical result is that every weight module of $W_{\ell }\left({\mathrm{sl}}_{2|1}\right)$ is $C_1$-cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on $L_k\left({\mathrm{sl}}_2\right)-wtmod$ for any admissible level $k$. As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that $L_k\left({\mathrm{sl}}_2\right)-wtmod$ is a ribbon category if and only if $L_{k+1}\left(g\right)-wtmod$ is, in particular, it follows that for admissible levels $k=-2+\frac{u}{v}$ and $v\in \{2,3\}$ and $u=-1\mathrm{mod}v$, the category $L_k\left({\mathrm{sl}}_2\right)-wtmod$ is a ribbon category.