{
  "id": "2504.10305",
  "version": "v3",
  "published": "2025-04-14T15:13:49.000Z",
  "updated": "2025-05-13T13:24:05.000Z",
  "title": "The commutator subalgebra of the Lie algebra associated with a right-angled Coxeter group",
  "authors": [
    "Fedor Vylegzhanin",
    "Yakov Veryovkin"
  ],
  "comment": "18 pages. v3: better exposition in section 6; submitted",
  "categories": [
    "math.GR",
    "math.AT"
  ],
  "abstract": "We study the graded Lie algebra $L(RC_K)$ associated with the lower central series of a right-angled Coxeter group $RC_K$. We prove that its commutator subalgebra is a quotient of the polynomial ring over an auxiliary Lie subalgebra $N_K$ of the graph Lie algebra $L_K$, and conjecture that the quotient map is an isomorphism. The epimorphism is defined in terms of a new operation in the associated Lie algebra, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra $U(N_K)$ is the mod 2 loop homology algebra of the corresponding moment-angle complex $Z_K$. This allows us to give a presentation of the Lie algebra $N_K$ by generators and relations.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2025-05-13T13:24:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F14",
      "20F12",
      "20F55",
      "57S12"
    ],
    "keywords": [
      "right-angled coxeter group",
      "commutator subalgebra",
      "lower central series",
      "loop homology algebra",
      "auxiliary lie subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}