{
  "id": "2212.04946",
  "version": "v1",
  "published": "2022-12-09T15:57:58.000Z",
  "updated": "2022-12-09T15:57:58.000Z",
  "title": "A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras",
  "authors": [
    "Yunhe Sheng"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2008.06714",
  "journal": "Bulletin London Math. Soc. (2022)",
  "doi": "10.1112/blms.12712",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA",
    "math.RA"
  ],
  "abstract": "In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\\infty$-algebra that controls deformations of a relative \\RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\\infty$-algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative Rota-Baxter Lie algebras with close connection to pre-Lie$_\\infty$-algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras of nonzero weights.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-09T15:57:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cohomology",
      "homotopy theories",
      "review deformation",
      "homotopy relative rota-baxter lie algebras",
      "voronovs higher derived brackets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}