{
  "id": "2012.04014",
  "version": "v1",
  "published": "2020-12-07T19:33:58.000Z",
  "updated": "2020-12-07T19:33:58.000Z",
  "title": "Reductive subalgebras of semisimple Lie algebras and Poisson commutativity",
  "authors": [
    "Dmitri I. Panyushev",
    "Oksana S. Yakimova"
  ],
  "categories": [
    "math.RT",
    "math.SG"
  ],
  "abstract": "Let $\\mathfrak g$ be a semisimple Lie algebra, $\\mathfrak h\\subset\\mathfrak g$ a reductive subalgebra such that $\\mathfrak h^\\perp$ is a complementary $\\mathfrak h$-submodule of $\\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\\mathcal S}(\\mathfrak g)$ by taking the subalgebra ${\\mathcal Z}$ generated by the bi-homogeneous components of all $H\\in{\\mathcal S}(\\mathfrak g)^{\\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\\mathcal Z}$. As a by-product, we prove that ${\\mathcal Z}$ is Poisson commutative if $\\mathfrak h$ is abelian and describe ${\\mathcal Z}$ in the special case when $\\mathfrak h$ is a Cartan subalgebra. In this case, ${\\mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(\\mathrm{dim}\\,\\mathfrak g+\\mathrm{rk}\\,\\mathfrak g)/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-07T19:33:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B63",
      "14L30",
      "17B08",
      "17B20",
      "22E46"
    ],
    "keywords": [
      "semisimple lie algebra",
      "poisson commutativity",
      "reductive subalgebra",
      "maximal transcendence degree",
      "symmetric algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}