The nonsemisimple Kazhdan–Lusztig category for affine sl2$\mathfrak {sl}_2$ at admissible levels

Abstract We show that Kazhdan and Lusztig's category of modules for the affine Lie algebra at an admissible level , equivalently the category of finite‐length grading‐restricted generalized modules for the universal affine vertex operator algebra , is a braided tensor category. Although this tensor category is not rigid, we show that the subcategory of all rigid objects in is equal to the subcategory of all projective objects, and that every simple module in has a projective cover. Moreover, we show that the full subcategory of projective objects in is monoidal equivalent to the category of tilting modules for quantum at the root of unity . Using this, we establish a universal property of the tensor category , 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 to the category of finite‐dimensional weight modules for the quantum group associated to at the root of unity . We also use the universal property to classify the categories 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 to a category of modules for the Virasoro algebra at central charge .