{
  "id": "2311.18594",
  "version": "v1",
  "published": "2023-11-30T14:37:16.000Z",
  "updated": "2023-11-30T14:37:16.000Z",
  "title": "Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads",
  "authors": [
    "Vladimir Dotsenko"
  ],
  "comment": "35 pages, comments are welcome",
  "categories": [
    "math.AT",
    "math.CT",
    "math.KT"
  ],
  "abstract": "We present a new equivalent definition of wheeled operads, and a related way to think of divergence of derivations of free operadic algebras, and use our viewpoint to study stable homology of Lie algebras of derivations of free $\\mathcal{O}$-algebras, for any augmented operad $\\mathcal{O}$. First, we relate stable homology of the positive part of the Lie algebra of all derivations to the homology of wheeled bar construction of $\\mathcal{O}$ with its trivial wheeled operad structure. This generalizes both the Loday-Quillen-Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of Lie algebra of vector fields with tensor coefficients. Second, we prove a \"nicer\" version of the latter stability theorem for the positive part of the Lie algebra of all derivations with zero divergence, relating its homology to the homology of the wheeled bar construction of the wheeled completion of the operad $\\mathcal{O}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-30T14:37:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "derivations",
      "homotopy invariants",
      "wheeled bar construction",
      "positive part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}