The non-semisimple Kazhdan-Lusztig category for affine $\mathfrak{sl}_2$ at admissible levels

We show that Kazhdan and Lusztig's category $KL^k(\mathfrak{sl}_2)$ of modules for the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ at an admissible level $k$, equivalently the category of finite-length grading-restricted generalized modules for the universal affine vertex operator algebra $V^k(\mathfrak{sl}_2)$, is a braided tensor category. Although this tensor category is not rigid, we show that the subcategory of all rigid objects in $KL^k(\mathfrak{sl}_2)$ is equal to the subcategory of all projective objects, and that every simple module in $KL^k(\mathfrak{sl}_2)$ has a projective cover. Moreover, we show that the full subcategory of projective objects in $KL^k(\mathfrak{sl}_2)$ is monoidal equivalent to the category of tilting modules for quantum $\mathfrak{sl}_2$ at the root of unity $\zeta=e^{\pi i/(k+2)}$. Using this, we establish a universal property of the tensor category $KL^k(\mathfrak{sl}_2)$, and as an application, we prove a weak Kazhdan-Lusztig correspondence, that is, we obtain an exact essentially surjective (but not full or faithful) tensor functor from $KL^k(\mathfrak{sl}_2)$ to the category of finite dimensional weight modules for the quantum group associated to $\mathfrak{sl}_2$ at the root of unity $\zeta $. We also use the universal property to classify the categories $KL^k(\mathfrak{sl}_2)$ up to (braided) tensor equivalence and to obtain a tensor-categorical version of quantum Drinfeld-Sokolov reduction, that is, we construct a braided tensor functor from $KL^k(\mathfrak{sl}_2)$ to a category of modules for the Virasoro algebra at central charge $1-\frac{6(k+1)^2}{k+2}$.