{
  "id": "1702.08011",
  "version": "v1",
  "published": "2017-02-26T09:47:06.000Z",
  "updated": "2017-02-26T09:47:06.000Z",
  "title": "Weak composition quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras",
  "authors": [
    "Li Guo",
    "Jean-Yves Thibon",
    "Houyi Yu"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra WCQSym of weak composition quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C.~Rota relating symmetric type functions and Rota-Baxter algebras. We provide the transformation formulas between the weak composition monomial and fundamental quasi-symmetric functions, which extends the corresponding results for quasi-symmetric functions. Moreover, we show that QSym is a Hopf subalgebra and a Hopf quotient algebra of WCQSym. Rota's question is addressed by identifying WCQsym with the free commutative unitary Rota-Baxter algebra of weight 1 on one generator, which also allows us to equip this algebra with a Hopf algebra structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-26T09:47:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "16W99",
      "16T33"
    ],
    "keywords": [
      "weak composition quasi-symmetric functions",
      "free commutative unitary rota-baxter algebra",
      "hopf algebra structure",
      "relating symmetric type functions",
      "hopf algebra qsym"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}