engleski

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

We study the semisimplicity of the category $KL_k$ for affine Lie superalgebras and provide a super analog of certain results from https://doi.org/10.1093/imrn/rny237. Let $\sKlf$ be the subcategory of $KL_k$ consisting of ordinary modules on which a Cartan subalgebra acts semisimply. We prove that $\sKlf$ is semisimple when 1) $k$ is a collapsing level, 2) $W_k(\g, \theta)$ is rational, 3) $W_k(\g, \theta)$ is semisimple in a certain category. The analysis of the semisimplicity of $\Kl$ is subtler than in the Lie algebra case, since in super case $KL_k$ can contain indecomposable modules. We are able to prove that in many cases when $\sKlf$ is semisimple we indeed have $\sKlf=\Kl$, which therefore excludes indecomposable and logarithmic modules in $\Kl$. In these cases we are able to prove that there is a conformal embedding $W \hookrightarrow V_k(\g)$ with $W$ semisimple (see Section \ref{; ; ; ; 9}; ; ; ; ). In particular, we prove the semisimplicity of $\Kl$ for $\g=sl(2\vert 1)$ and $k = -\frac{; ; ; ; m+1}; ; ; ; {; ; ; ; m+2}; ; ; ; $, $m \in {; ; ; ; \Z}; ; ; ; _{; ; ; ; \ge 0}; ; ; ; $. For $\g =sl(m \vert 1)$, we prove that $\Kl$ 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 $\sKlf$.

vertex algebras, affine superalgebras, category KL.

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