{
  "id": "1905.09630",
  "version": "v1",
  "published": "2019-05-23T13:06:00.000Z",
  "updated": "2019-05-23T13:06:00.000Z",
  "title": "Lie algebras of differential operators III: Classification",
  "authors": [
    "Helge Øystein Maakestad"
  ],
  "categories": [
    "math.AG",
    "math.KT",
    "math.RA",
    "math.RT"
  ],
  "abstract": "In a previous paper we introduced the notion of a D-Lie algebra $\\tilde{L}$. A D-Lie algebra $\\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-cocycle $f\\in \\operatorname{Z}^2(\\operatorname{Der}_k(A),A)$ a functor $F_{f}(-)$ from the category of $A/k$-Lie-Rinehart algebras to the category of D-Lie algebras and classify D-Lie algebras with projective canoncial quotient using the functor $F_{f}(-)$. We prove a similar classification for non-abelian extensions of D-Lie algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-23T13:06:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S30",
      "17B45",
      "17B56",
      "17B35",
      "20G10",
      "20G05"
    ],
    "keywords": [
      "differential operators",
      "classify d-lie algebras",
      "classification",
      "lie-rinehart algebra",
      "non-abelian extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}