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.
Categories: math.QA hep-th math.RT