{
  "id": "2403.14959",
  "version": "v1",
  "published": "2024-03-22T05:27:23.000Z",
  "updated": "2024-03-22T05:27:23.000Z",
  "title": "Reducibility of commuting varieties of elements of simple Lie algebra",
  "authors": [
    "Nikola Kovačević"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper, we prove that the variety $C_m(L)$ of commuting $m$-tuples of elements of simple Lie algebra $L$ is often reducible. Explicitely, we prove it is reducible for all simple Lie algebra $L$ not isomorphic to $\\mathfrak{sl}_2$ and $\\mathfrak{sl})_3$, and all $m \\geq 4$. We also prove it is reducible for $C_3(L)$ for $L$ of types $B_k,C_k,E_7,E_8,F_4,G_2$, $k \\geq 2$, as well as for $D_l$ for $l \\geq 10$. We do this by proving Theorem on Adding Diagonals, that says that if we can find a simple Lie subalgebra $L'$ whose Dynkin diagram is a subdiagram of the Dynkin diagram of $L$, then under mild conditions, from the fact that $C_m(L')$ is reducible, it follows that $C_m(L)$ is also reducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-22T05:27:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lie algebra",
      "commuting varieties",
      "reducibility",
      "dynkin diagram",
      "simple lie subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}