Tensor Structure on the Kazhdan–Lusztig Category for Affine 𝔤𝔩(1|1)

Abstract

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.