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

Drazen Adamovic; Pierluigi Moseneder Frajria; Paolo Papi

doi:10.1016/j.aim.2022.108493

paper

On the semisimplicity of the category $K L_{k}$ for affine Lie superalgebras

arXiv:2107.12105 · doi:10.1016/j.aim.2022.108493

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 $K L_{k}^{f in}$ be the subcategory of $K L_{k}$ consisting of ordinary modules on which the Cartan subalgebra acts semisimply. We prove that $K L_{k}^{f in}$ is semisimple when 1) $k$ is a collapsing level, 2) $W_{k} (g, θ)$ is rational, 3) $W_{k} (g, θ)$ 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 $K L_{k}^{f in}$ is semisimple we indeed have $K L_{k}^{f in} = K L_{k}$ , 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 $W ↪ V_{k} (g)$ with $W$ semisimple (see Section 10). In particular, we prove the semisimplicity of $K L_{k}$ for $g = s l (2∣1)$ and $k = - \frac{m + 1}{m + 2}$ , $m \in Z_{\geq 0}$ . For $g = s l (m ∣1)$ , 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 $K L_{k}^{f in}$ .

arXiv comment28 pages, latex file, to appear in Advances in Mathematics

References in corpus (11)

Cited by in corpus (1)

On the structure of W-algebras in type A

On the semisimplicity of the category KLk​ for affine Lie superalgebras

Abstract

References in corpus (11)

Cited by in corpus (1)

On the semisimplicity of the category $K L_{k}$ for affine Lie superalgebras