On the semisimplicity of the category KLk 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 KLfin be the subcategory of KLk consisting k of ordinary modules on which a Cartan subalgebra acts semisimply. We prove that KLfin is semisimple when 1) k 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 KLfin is semisimple we indeed have KLfin = KLk, which kk therefore excludes indecomposable and logarithmic modules in KLk. In these cases we are able to prove that there is a conformal embedding W → V_k(g) with W semisimple (see Section 10). In particular, we prove the semisimplicity of KLk for g = sl(2|1) and k = −(m+1)/(m+2) , 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 KLfin