{
  "id": "1204.5395",
  "version": "v1",
  "published": "2012-04-24T14:56:02.000Z",
  "updated": "2012-04-24T14:56:02.000Z",
  "title": "On the Hall algebra of semigroup representations over F_1",
  "authors": [
    "Matt Szczesny"
  ],
  "categories": [
    "math.RT",
    "math.CO",
    "math.CT",
    "math.RA"
  ],
  "abstract": "Let $\\A$ be a finitely generated semigroup with 0. An $\\A$-module over $\\fun$ (also called an $\\A$--set), is a pointed set $(M,*)$ together with an action of $\\A$. We define and study the Hall algebra $\\H_{\\A}$ of the category $\\C_{\\A}$ of finite $\\A$--modules. $\\H_{\\A}$ is shown to be the universal enveloping algebra of a Lie algebra $\\n_{\\A}$, called the \\emph{Hall Lie algebra} of $\\C_{\\A}$. In the case of the $\\fm$ - the free monoid on one generator $\\fm$, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\\fm$-modules) is isomorphic to Kreimer's Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\\A$ is a quotient of $\\fm$ by a congruence, and the monoid $G \\cup \\{0\\}$ for a finite group $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-24T14:56:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hall algebra",
      "semigroup representations",
      "lie algebra",
      "kreimers hopf algebra",
      "rooted forests"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.5395S"
    }
  }
}