{
  "id": "1905.03020",
  "version": "v1",
  "published": "2019-05-08T12:02:43.000Z",
  "updated": "2019-05-08T12:02:43.000Z",
  "title": "On the Adjoint Representation of a Hopf Algebra",
  "authors": [
    "Martin Lorenz",
    "Bach Nguyen",
    "Ramy Yammine"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For cocommutative $H$, we show that $H_{\\text{adfin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any pointed Hopf algebra. If $H$ is generated by grouplike and by skew-primitive elements, then $H_{\\text{adfin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-08T12:02:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16T05",
      "16T20"
    ],
    "keywords": [
      "adjoint representation",
      "locally finite parts yield",
      "left coideal subalgebra",
      "group theory",
      "finite-dimensional subrepresentations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}