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

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