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

The category of weight modules $L_k(\mathfrak{sl}_2)\text{-wtmod}$ of the simple affine vertex algebra of $\mathfrak{sl}_2$ 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 $W_\ell(\mathfrak{sl}_{2|1})$, the simple prinicipal $W$-algebra of $\mathfrak{sl}_{2|1}$ at level $\ell$ (the $N=2$ super conformal algebra) where the levels are related via $(k+2)(\ell+1)=1$. Every weight module of $W_\ell(\mathfrak{sl}_{2|1})$ is lower-bounded and has finite-dimensional conformal weight spaces. The main technical result is that every weight module of $W_\ell(\mathfrak{sl}_{2|1})$ 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 $L_k(\mathfrak{sl}_2)\text{-wtmod}$ for any admissible level $k$. As applications, the fusion rules of ordinary modules with any weight module are computed and it is shown that $L_k(\mathfrak{sl}_2)\text{-wtmod}$ is a ribbon category if and only if $L_{k+1}(\mathfrak{sl}_2)\text{-wtmod}$ is, in particular it follows that for admissible levels $k = - 2 + \frac{u}{v}$ and $v \in \{2, 3\}$ and $u = -1 \mod v$ the category $L_k(\mathfrak{sl}_2)\text{-wtmod}$ is a ribbon category.