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

TL;DR

This paper establishes a vertex algebraic braided tensor supercategory structure for the Kazhdan-Lusztig category of affine l(1|1), analyzes its subcategory of semisimple actions, and explores applications to related affine Lie superalgebras.

Contribution

It introduces a tensor supercategory structure on the Kazhdan-Lusztig category for l(1|1) and determines fusion rules, projective covers, and rigidity, with applications to l(2|1).

Findings

Established braided tensor supercategory structure for KL_k.

Determined all fusion rules involving simple and projective objects.

Proved rigidity of the categories using KZ equations.

Abstract

We show that the Kazhdan-Lusztig category $KL_k$ 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 $KL_k$ 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 $KL_k$ 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.