{
  "id": "2008.11179",
  "version": "v1",
  "published": "2020-08-25T17:23:58.000Z",
  "updated": "2020-08-25T17:23:58.000Z",
  "title": "Universal tensor categories generated by dual pairs",
  "authors": [
    "Alexandru Chirvasitu",
    "Ivan Penkov"
  ],
  "comment": "35 pages + references",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "abstract": "Let $V_*\\otimes V\\rightarrow\\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\\mathfrak{g}=\\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $\\varphi$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $\\varphi^*: V^*\\rightarrow V^*$. We study the tensor Grothendieck category $\\mathbb{T}$ generated by the $\\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\\mathbb{C}$ is no longer injective in $\\mathbb{T}$. We describe the injective hull $I$ of $\\mathbb{C}$ in $\\mathbb{T}$, and show that the category $\\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\\mathbb{T}$ of objects in $\\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\\mathbb{T}$ is also Koszul, and moreover that ${}_I\\mathbb{T}$ is universal among abelian $\\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\\hookrightarrow X$, $Y'\\hookrightarrow Y$ and a pairing $X\\otimes Y\\rightarrow \\text{\\textbf{1}}$ where \\textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\\mathbb{T}$ and ${}_I\\mathbb{T}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-25T17:23:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B10",
      "18M05",
      "18E10",
      "16T15",
      "16S37"
    ],
    "keywords": [
      "universal tensor categories",
      "dual pairs",
      "countable-dimensional complex vector spaces",
      "mackey lie algebra",
      "tensor grothendieck category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}