{
  "id": "1903.01226",
  "version": "v1",
  "published": "2019-03-04T13:29:13.000Z",
  "updated": "2019-03-04T13:29:13.000Z",
  "title": "Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra",
  "authors": [
    "Samuel A. Lopes",
    "Andrea Solotar"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For each nonzero $h\\in \\mathbb{F}[x]$, where $\\mathbb{F}$ is a field, let $\\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description the Hochschild cohomology $\\mathsf{HH}^\\bullet(\\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\\mathbb{F}$ has positive characteristic, the center of $\\mathsf{A}_h$ is nontrivial and we describe $\\mathsf{HH}^\\bullet(\\mathsf{A}_h)$ as a module over its center. The most interesting results occur when $\\mathbb{F}$ has characteristic $0$. In this case, we describe $\\mathsf{HH}^\\bullet(\\mathsf{A}_h)$ as a module over the Lie algebra $\\mathsf{HH}^1(\\mathsf{A}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\\mathsf{HH}^\\bullet(\\mathsf{A}_h)$ is a semisimple $\\mathsf{HH}^1(\\mathsf{A})$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-04T13:29:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weyl algebra",
      "hochschild cohomology",
      "lie structure",
      "subalgebras",
      "intermediate series modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}