Tensor categories of affine Lie algebras beyond admissible levels

Abstract

We show that if $V$ is a vertex operator algebra such that all the irreducible ordinary $V$-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for $V$ are of finite length, then the category of finite length generalized $V$-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $\mathfrak{g}$ at level $k$ and the category $K{L}_k(\mathfrak{g})$ of its finite length generalized modules, we discover several families of $K{L}_k(\mathfrak{g})$ at non-admissible levels $k$, having braided tensor category structures. In particular, $K{L}_k(\mathfrak{g})$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $K{L}_k(\mathfrak{g})$, including the category $K{L}_{-1}({\mathfrak{sl}}_n)$. Using these results, we construct a rigid tensor category structure on a full subcategory of $K{L}_1(\mathfrak{sl}(n|m))$ consisting of objects with semisimple Cartan subalgebra actions.

arXiv commentWe add more details to the proofs and also incorporate some recent progress mainly in Sec. 4 and Sec. 6, we also add Sec. 2.4 summarizing the main results on tensor category theory of vertex (super)algebra extensions that are used in this paper

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Cited by in corpus (16)

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• On ribbon categories for singlet vertex algebras
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• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• $N=4$ superconformal algebras and diagonal cosets
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras
• A general mirror equivalence theorem for coset vertex operator algebras
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• $N=1$ super Virasoro tensor categories
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras