{
  "id": "2105.09731",
  "version": "v1",
  "published": "2021-05-20T13:31:38.000Z",
  "updated": "2021-05-20T13:31:38.000Z",
  "title": "Free limits of free algebras",
  "authors": [
    "Alexandru Chirvasitu",
    "Tao Hong"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RA",
    "math.CT"
  ],
  "abstract": "Consider a diagram $\\cdots \\to F_3 \\to F_2\\to F_1$ of algebraic systems, where $F_n$ denotes the free object on $n$ generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the $F_i$ are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the $F_i$ are free Lie algebras then the limit in the category of graded Lie algebras is again free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T13:31:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "08B20",
      "16S10",
      "16W50",
      "17B01",
      "17B70"
    ],
    "keywords": [
      "free algebras",
      "free limits",
      "free formal power series algebra",
      "free lie algebras",
      "distinguished trivial element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}