Braided tensor categories of admissible modules for affine Lie algebras

Abstract

Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is ${\widehat{\mathfrak{sl}}}_2$, we prove the rigidity and modularity conjectures.

Comments are welcome

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Cited by in corpus (34)

• Trialities of $\mathcal{W}$-algebras
• On fusion rules and intertwining operators for the Weyl vertex algebra
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• On ribbon categories for singlet vertex algebras
• Tensor categories of affine Lie algebras beyond admissible levels
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Rectangular W-algebras, extended higher spin gravity and dual coset CFTs
• Direct limit completions of vertex tensor categories
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Unitary and non-unitary $N=2$ minimal models
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• 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 $A$
• Generalized parafermions of orthogonal type
• A general mirror equivalence theorem for coset vertex operator algebras
• Full Logarithmic Conformal Field theory - an Attempt at a Status Report
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• Tensor Categories arising from the Virasoro Algebra
• Ito's conjecture and the coset construction for ${\mathcal{W}}^k(\mathfrak{sl}(3|2))$
• $N=1$ super Virasoro tensor categories
• Staggered modules of $N=2$ superconformal minimal models
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Orthosymplectic Feigin-Semikhatov duality
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• $A_{2l}^{(2)}$ at level $-l-\frac{1}{2}$
• Vertex Algebras and Commutative Algebras
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Representation theory of vertex operator algebras and orbifold conformal field theory
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra