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

Abstract

We show that Kazhdan and Lusztig's category $K{L}^k({\mathfrak{sl}}_2)$ 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 $K{L}^k({\mathfrak{sl}}_2)$ is equal to the subcategory of all projective objects, and that every simple module in $K{L}^k({\mathfrak{sl}}_2)$ has a projective cover. Moreover, we show that the full subcategory of projective objects in $K{L}^k({\mathfrak{sl}}_2)$ 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 $K{L}^k({\mathfrak{sl}}_2)$, 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 $K{L}^k({\mathfrak{sl}}_2)$ 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 $K{L}^k({\mathfrak{sl}}_2)$ 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 $K{L}^k({\mathfrak{sl}}_2)$ to a category of modules for the Virasoro algebra at central charge $1-\frac{6(k+1{)}^2}{k+2}$.

73 pages, final version incorporating referee comments and added references, appendix on structure of tilting modules removed and replaced with references to the quantum group literature

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Cited by in corpus (1)

• Rigidity of non-negligible objects of moderate growth in braided categories