{
  "id": "1011.1536",
  "version": "v1",
  "published": "2010-11-06T04:24:54.000Z",
  "updated": "2010-11-06T04:24:54.000Z",
  "title": "Polytopes, Hopf algebras and Quasi-symmetric functions",
  "authors": [
    "Victor M. Buchstaber",
    "Nickolai Erokhovets"
  ],
  "comment": "61 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product $\\times $ and a join $\\divideontimes$ of polytopes. $(\\mathcal{P},\\times)$ is a commutative associative bigraded ring of polynomials, and $\\mathcal{RP}=(\\mathbb Z\\varnothing\\oplus\\mathcal{P},\\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\\mathcal{RP}$ has the structure of a graded Hopf algebra. It turns out that $\\mathcal{P}$ has a natural Hopf comodule structure over $\\mathcal{RP}$. Faces operators $d_k$ that send a polytope to the sum of all its $(n-k)$-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra $\\mathcal{Z}$. This structure gives a ring homomorphism $\\R\\to\\Qs\\otimes\\R$, where $\\R$ is $\\mathcal{P}$ or $\\mathcal{RP}$. Composing this homomorphism with the characters $P^n\\to\\alpha^n$ of $\\mathcal{P}$, $P^n\\to\\alpha^{n+1}$ of $\\mathcal{RP}$, and with the counit we obtain the ring homomorphisms $f\\colon\\mathcal{P}\\to\\Qs[\\alpha]$, $f_{\\mathcal{RP}}\\colon\\mathcal{RP}\\to\\Qs[\\alpha]$, and $\\F^*:\\mathcal{RP}\\to\\Qs$, where $F$ is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over $\\mathbb Q$, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism $f,\\;f_{\\mathcal{RP}}$, and $\\F$ the images of two polytopes coincide if and only if they have equal flag $f$-vectors. Therefore algebraic structures on the images give the information about flag $f$-vectors of polytopes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-06T04:24:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52B05",
      "16T05",
      "57T25"
    ],
    "keywords": [
      "quasi-symmetric functions",
      "combinatorial polytopes",
      "natural hopf comodule structure",
      "polynomials",
      "dimensional faces define"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1070/RM2011v066n02ABEH004741",
      "journal": "Russian Mathematical Surveys",
      "year": 2011,
      "month": "Apr",
      "volume": 66,
      "number": 2,
      "pages": 271
    },
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011RuMaS..66..271B"
    }
  }
}