{
  "id": "2406.14800",
  "version": "v1",
  "published": "2024-06-21T00:34:50.000Z",
  "updated": "2024-06-21T00:34:50.000Z",
  "title": "Multi-quasisymmetric functions with semigroup exponents, Hopf algebras and Rota-Baxter algebras",
  "authors": [
    "Xing Gao",
    "Li Guo",
    "Xiao-Song Peng"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO",
    "math.AC",
    "math.RA"
  ],
  "abstract": "Many years ago, G.-C.~Rota discovered a close connection between symmetric functions and Rota-Baxter algebras, and proposed to study generalizations of symmetric functions in the framework of Rota-Baxter algebras. Guided by this proposal, quasisymmetric functions from weak composition (instead of just compositions) were obtained from free Rota-Baxter algebras on one generator. This paper aims to generalize this approach to free Rota-Baxter algebras on multiple generators in order to obtain further generalizations of quasisymmetric functions. For this purpose and also for its independent interest, the space $\\mathrm{MQSym}$ of quasisymmetric functions on multiple sequences of variables is defined, generalizing quasisymmetric functions and diagonally quasisymmetric functions of Aval, Bergeron and Bergeron. Linear bases of such multi-quasisymmetric functions are given by monomial multi-quasisymmetric functions and fundamental multi-quasisymmetric functions, the latter recover the fundamental $G^m$-quasisymmetric functions of Aval and Chapoton. Next introduced is the even more general notion of multi-quasisymmetric functions $\\mathrm{MQSym}^E$ with exponents in a semigroup $E$, which also generalizes the quasisymmetric functions with semigroup exponents in a recent work. Through this approach, a natural Hopf algebraic structure is obtained on $\\mathrm{MQSym}^E$. Finally, in support of Rota's proposal, the free commutative unitary Rota-Baxter algebra on a finite set is shown to be isomorphic to a scalar extension of $\\mathrm{MQSym}^E$, a fact which in turn equips the free Rota-Baxter algebra with a Hopf algebra structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-21T00:34:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "16W99",
      "17B38",
      "08B20",
      "16T30"
    ],
    "keywords": [
      "multi-quasisymmetric functions",
      "semigroup exponents",
      "free rota-baxter algebra",
      "natural hopf algebraic structure",
      "free commutative unitary rota-baxter algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}