{
  "id": "2011.13002",
  "version": "v1",
  "published": "2020-11-25T20:08:41.000Z",
  "updated": "2020-11-25T20:08:41.000Z",
  "title": "Unstable algebras over an operad II",
  "authors": [
    "Sacha Ikonicoff"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra in characteristic $p>2$. We first study algebras over an operad in the category of graded vector space with Frobenius map. Under suitable conditions, we show that the free $\\mathcal{P}$-algebra with Frobenius map compatible to the action of $\\mathcal P$ over a graded vector space $V$ with free Frobenius map is itself a free $\\mathcal{P}$-algebra generated by $V$. We then define $\\star$-unstable $\\mathcal{P}$-algebras over the Steenrod algebra, where $\\mathcal{P}$ is an operad and $\\star$ is a completely symmetric $p$-ary operation in $\\mathcal{P}$. Under some hypotheses on $\\star$ and on the unstable module $M$, we identify the free $\\star$-unstable $\\mathcal{P}$-algebra generated by $M$ as a free $\\mathcal{P}$-algebra. We show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, and Brown and Gitler, that takes into account their internal product. Finally, we generalise a result due to Campbell and Selick which shows that we can twist the action of the Steenrod algebra on the free unstable algebra generated by a direct sum of unstable modules without modifying the underlying unstable module structure, and we obtain a decomposition for some of our new constructions which involve the Carlsson modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-25T20:08:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55S10",
      "18D50",
      "17A30"
    ],
    "keywords": [
      "unstable algebra",
      "steenrod algebra",
      "graded vector space",
      "free frobenius map",
      "first study algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}