{
  "id": "2107.12105",
  "version": "v1",
  "published": "2021-07-26T10:52:47.000Z",
  "updated": "2021-07-26T10:52:47.000Z",
  "title": "On the semisimplicity of the category $KL_k$ for affine Lie superalgebras",
  "authors": [
    "Drazen Adamovic",
    "Pierluigi Moseneder Frajria",
    "Paolo Papi"
  ],
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "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 $KL_k^{fin}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-26T10:52:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine lie superalgebras",
      "semisimplicity",
      "semisimple",
      "cartan subalgebra acts",
      "lie algebra case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}