On the semisimplicity of the category $KL_k$ for affine Lie superalgebras

We study the semisimplicity of the category KLk for affine Lie superalgebras and provide a super analog of certain results from [8]. Let KLk^fin be the subcategory of KLk consisting of ordinary modules on which a Cartan subalgebra acts semisimply. We prove that KLk^fin is semisimple when 1) k is a collapsing level, 2) Wk(g,θ) is rational, 3) Wk(g,θ) is semisimple in a certain category. The analysis of the semisimplicity of KLk is subtler than in the Lie algebra case, since in super case KLk can contain indecomposable modules. We are able to prove that in many cases when KLk^fin is semisimple we indeed have KLk^fin=KLk, which therefore excludes indecomposable and logarithmic modules in KLk. In these cases we are able to prove that there is a conformal embedding W↪Vk(g) with W semisimple (see Section 10). In particular, we prove the semisimplicity of KLk for g=sl(2|1) and [Formula presented], m∈Z≥0. For g=sl(m|1), we prove that KLk is semisimple for k=−1, but for k a positive integer we show that it is not semisimple by constructing indecomposable highest weight modules in KLk^fin.