{
  "id": "2411.12912",
  "version": "v1",
  "published": "2024-11-19T22:57:22.000Z",
  "updated": "2024-11-19T22:57:22.000Z",
  "title": "Triangular decompositions: Reedy algebras and quasi-hereditary algebras",
  "authors": [
    "Teresa Conde",
    "Georgios Dalezios",
    "Steffen Koenig"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Finite-dimensional Reedy algebras form a ring-theoretic analogue of Reedy categories and were recently proved to be quasi-hereditary. We identify Reedy algebras as quasi-hereditary algebras admitting a triangular (or Poincar\\'e-Birkhoff-Witt type) decomposition into the tensor product of two oppositely directed subalgebras over a common semisimple subalgebra. This exhibits homological and representation-theoretic structure of the ingredients of the Reedy decomposition and it allows to give a characterisation of Reedy algebras in terms of idempotent ideals occurring in heredity chains, providing an analogue for Reedy algebras of a result of Dlab and Ringel on quasi-hereditary algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-19T22:57:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasi-hereditary algebras",
      "triangular decompositions",
      "finite-dimensional reedy algebras form",
      "common semisimple subalgebra",
      "idempotent ideals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}