{
  "id": "1512.07944",
  "version": "v1",
  "published": "2015-12-25T02:25:29.000Z",
  "updated": "2015-12-25T02:25:29.000Z",
  "title": "Graphs and Metric 2-step Nilpotent Lie Algebras",
  "authors": [
    "Rachelle DeCoste",
    "Lisa DeMeyer",
    "Meera Mainkar"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra $\\mathfrak{n}_G$ from a simple directed graph $G$ in 2005. There is a natural inner product on $\\mathfrak{n}_G$ arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group $N$ with Lie algebra $\\mathfrak{n}_G$. We classify singularity properties of the Lie algebra $\\mathfrak{n}_G$ in terms of the graph $G$. A comprehensive description is given of graphs $G$ which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph $G$ and on a lattice $\\Gamma \\subseteq N$ for which the quotient $\\Gamma \\backslash N$, a compact nilmanifold, has a dense set of smoothly closed geodesics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-25T02:25:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E25",
      "53C30",
      "53C22"
    ],
    "keywords": [
      "nilpotent lie algebra",
      "nilpotent lie group",
      "natural inner product",
      "study geometric properties",
      "simple directed graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}