{
  "id": "2309.02522",
  "version": "v1",
  "published": "2023-09-05T18:24:54.000Z",
  "updated": "2023-09-05T18:24:54.000Z",
  "title": "Representations of large Mackey Lie algebras and universal tensor categories",
  "authors": [
    "Ivan Penkov",
    "Valdemar Tsanov"
  ],
  "comment": "41 pages",
  "categories": [
    "math.RT",
    "math.CO",
    "math.CT",
    "math.RA"
  ],
  "abstract": "We extend previous work by constructing a universal abelian tensor category ${\\bf T}_t$ generated by two objects $X,Y$ equipped with finite filtrations $0\\subsetneq X_0\\subsetneq ... X_{t+1}\\subsetneq X$ and $0\\subsetneq Y_0\\subsetneq ... Y_{t+1}\\subsetneq Y$, and with a pairing $X\\otimes Y\\to \\mathbb{I}$, where $\\mathbb{I}$ is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra $\\mathfrak{gl}^M(V,V_*)$ of cardinality $2^{\\aleph_t}$, associated to a diagonalizable pairing between two complex vector spaces $V,V_*$ of dimension $\\aleph_t$. As a preliminary step, we study a tensor category $\\mathbb{T}_t$ generated by the algebraic duals $V^*$, $(V_*)^*$. The injective hull of $\\mathbb{C}$ in $\\mathbb{T}_t$ is a commutative algebra $I$, and the category ${\\bf T}_t$ is consists of the free $I$-modules in $\\mathbb{T}_t$. An essential novelty in our work is the explicit computation of Ext-groups between simples in both categories ${\\bf T}_t$ and $\\mathbb{T}_t$, which had been an open problem already for $t=0$. This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-05T18:24:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B10",
      "18M05",
      "18E10",
      "16S37"
    ],
    "keywords": [
      "large mackey lie algebras",
      "universal tensor categories",
      "representations",
      "universal abelian tensor category",
      "complex vector spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}