On the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras

Abstract

We study the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras and provide a super analog of certain results from arXiv:1801.09880. Let $K{L}_k^{fin}$ be the subcategory of $K{L}_k$ consisting of ordinary modules on which the Cartan subalgebra acts semisimply. We prove that $K{L}_k^{fin}$ 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 $K{L}_k$ is subtler than in the Lie algebra case, since in super case $K{L}_k$ can contain indecomposable modules. We are able to prove that in many cases when $K{L}_k^{fin}$ is semisimple we indeed have $K{L}_k^{fin}=K{L}_k$, which therefore excludes indecomposable and logarithmic modules in $K{L}_k$. 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 $K{L}_k$ for $\mathfrak{g}=sl(2|1)$ and $k=-\frac{m+1}{m+2}$, $m\in {\mathbb{Z}}_{\ge 0}$. For $\mathfrak{g}=sl(m|1)$, we prove that $K{L}_k$ is semisimple for $k=-1$, but for $k=1$ we show that it is not semisimple by constructing indecomposable highest weight modules in $K{L}_k^{fin}$.

28 pages, latex file, to appear in Advances in Mathematics

References in corpus (1)

• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules

Cited by in corpus (1)

• On the structure of W-algebras in type A