{
  "id": "2307.06514",
  "version": "v1",
  "published": "2023-07-13T01:26:03.000Z",
  "updated": "2023-07-13T01:26:03.000Z",
  "title": "Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras",
  "authors": [
    "Daniil Klyuev"
  ],
  "comment": "42 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP",
    "math.QA"
  ],
  "abstract": "Let $\\mathcal{A}$ be a generalized Weyl or $q$-Weyl algebra, $M$ be an $\\mathcal{A}$-$\\overline{\\mathcal{A}}$ bimodule. Choosing an automorphism $\\rho$ of $\\mathcal{A}$ we can define the notion of an invariant Hermitian form: $(au,v)=(u,v\\rho(a))$ for all $a\\in \\mathcal{A}$ and $u,v\\in M$. Papers [P. Etingof, D. Klyuev, E. Rains, D. Stryker. Twisted traces and positive forms on quantized Kleinian singularities of type A. SIGMA 17 (2021), 029, 31 pages, arXiv:2009.09437] and [D. Klyuev. Twisted traces and positive forms on generalized q-Weyl algebras. SIGMA 18 (2022), 009, 28 pages, arXiv:2105.12652] obtained a classification of invariant positive definite Hermitian forms in the case when $M=\\mathcal{A}=\\overline{\\mathcal{A}}$, the case of the regular bimodule. We obtain a classification of invariant positive definite forms on $M$ in the case when $\\mathcal{A}$ has no finite-dimensional representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T01:26:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weyl algebra",
      "generalized weyl",
      "harish-chandra bimodules",
      "invariant positive definite hermitian forms",
      "unitarizability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}