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

TL;DR

This paper investigates the conditions under which the category of modules for affine Lie superalgebras is semisimple, extending known results and identifying cases with indecomposable modules, including explicit examples.

Contribution

It provides a super analog of previous semisimplicity results, characterizes when $KL_k^{fin}$ is semisimple, and identifies cases where $KL_k$ contains indecomposable modules.

Findings

$KL_k^{fin}$ is semisimple at collapsing levels and when associated $W$-algebras are rational or semisimple.

In many cases, $KL_k^{fin}=KL_k$, excluding indecomposable modules.

Proved semisimplicity for specific superalgebras like $sl(2|1)$ and $sl(m|1)$ at certain levels.

Abstract

We study the semisimplicity of the category $KL_k$ for affine Lie superalgebras and provide a super analog of certain results from arXiv:1801.09880. Let $KL_k^{fin}$ be the subcategory of $KL_k$ consisting of ordinary modules on which the Cartan subalgebra acts semisimply. We prove that $KL_k^{fin}$ is semisimple when 1) $k$ is a collapsing level, 2) $W_k(\mathfrak{g}, \theta)$ is rational, 3) $W_k(\mathfrak{g}, \theta)$ is semisimple in a certain category. The analysis of the semisimplicity of $KL_k$ 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 $KL_k^{fin}$ is semisimple we indeed have $KL_k^{fin}=KL_k$, which therefore excludes indecomposable and logarithmic modules in $KL_k$. In these cases we are able to prove that there is a conformal embedding $W \hookrightarrow V_k(\mathfrak{g})$ with $W$ semisimple (see Section 10). In particular, we prove the semisimplicity of $KL_k$ for $\mathfrak{g}=sl(2\vert 1)$ and $k = -\frac{m+1}{m+2}$, $m \in {\mathbb Z}_{\ge 0}$. For $\mathfrak{g} =sl(m \vert 1)$, we prove that $KL_k$ is semisimple for $k=-1$, but for $k=1$ we show that it is not semisimple by constructing indecomposable highest weight modules in $KL_k^{fin}$.