{
  "id": "1909.00443",
  "version": "v1",
  "published": "2019-09-01T18:32:34.000Z",
  "updated": "2019-09-01T18:32:34.000Z",
  "title": "Invariant theory and wheeled PROPs",
  "authors": [
    "Harm Derksen",
    "Visu Makam"
  ],
  "comment": "28 pages",
  "categories": [
    "math.RT",
    "math.AC",
    "math.RA"
  ],
  "abstract": "We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\\mathcal V}=T(V)\\otimes T(V^\\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector space over a field of $K$ of characteristic 0. First we classify all the ideals of the initial object ${\\mathcal{Z}}$ in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of ${\\mathcal V}$ are exactly subalgebras of the form ${\\mathcal V}^G$ where $G$ is a closed, reductive subgroup of the general linear group ${\\rm GL}(V)$. When $V$ is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the $n$-th Cayley-Hamilton identity can be embedded in an $n \\times n$ matrix ring over a commutative algebra $R$. Namely, we prove that a wheeled PROP can be embedded in $R\\otimes {\\mathcal V}$ for a commutative $K$-algebra $R$ if and only if it satisfies certain relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-01T18:32:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A72",
      "13A50",
      "18D50"
    ],
    "keywords": [
      "wheeled prop",
      "invariant theory",
      "tensor algebra",
      "finite dimensional hilbert space",
      "th cayley-hamilton identity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}