Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

Abstract

We show that the Kazhdan-Lusztig category $K{L}_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and that its full subcategory ${\mathcal{O}}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $K{L}_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik-Zamolodchikov equations, we prove that $K{L}_k$ and ${\mathcal{O}}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on ${\mathcal{O}}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

arXiv comment46 pages, to appear in Int. Math. Res. Not

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• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• $N=1$ 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
• The free field representation for the $GL(1|1)$ WZW model revisited
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras