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 $1-\frac{6(k+1{)}^2}{k+2}$.

arXiv comment73 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

References in corpus (26)

• Logarithmic extensions of minimal models: characters and modular transformations
• A universal characterization of higher algebraic K-theory
• Braided tensor categories and extensions of vertex operator algebras
• Tensor categories for vertex operator superalgebra extensions
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Fusion rules and boundary conditions in the c=0 triplet model
• Braided tensor categories of admissible modules for affine Lie algebras
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• A quasi-Hopf algebra for the triplet vertex operator algebra
• On the tensor structure of modules for compact orbifold vertex operator algebras
• Symplectic fermions and a quasi-Hopf algebra structure on ${\bar{U}}_{i}sl(2)$
• Gluing vertex algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• On ribbon categories for singlet vertex algebras
• Tensor categories of affine Lie algebras beyond admissible levels
• Diagram categories for ${\text{U}}_q$-tilting modules at roots of unity
• Direct limit completions of vertex tensor categories
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Braided zesting and its applications
• SL2 tilting modules in the mixed case
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$
• Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge

Cited by in corpus (1)

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