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

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