{
  "id": "2111.03378",
  "version": "v1",
  "published": "2021-11-05T10:36:48.000Z",
  "updated": "2021-11-05T10:36:48.000Z",
  "title": "Determinantal Ideals and the Canonical Commutation Relations. Classically or Quantized",
  "authors": [
    "Hans Plesner Jakobsen"
  ],
  "comment": "66 pages LaTeX",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA",
    "math.RT"
  ],
  "abstract": "We construct homomorphic images of $su(n,n)^{\\mathbb C}$ in Weyl Algebras ${\\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\\mathcal H}_{2nr}$, $su(n,n)^{\\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their list, we construct a representation of $u(r)$ in the Fock space commuting with $su(n,n)^{\\mathbb C}$, and this gives the multiplicities. The construction also gives an easy proof that the ideals of $(r+1)\\times (r+1)$ minors are prime ($r\\leq n-1)$. The quotients of all polynomials by such ideals carry the more singular of the representations. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly which representations from our list are missing some ${\\mathfrak k}^{\\mathbb C}$-types, thereby revealing exactly which covariant differential operators have unitary null spaces. We prove the analogous results for ${\\mathcal U}_q(su(n,n)^{\\mathbb C})$. The Weyl Algebras are replaced by the Hayashi--Weyl Algebras ${\\mathcal H}{\\mathcal W}_{2nr}$ and the Fock space by a $q$-Fock space. Further, determinants are replaced by $q$-determinants, and a commuting representation of ${\\mathcal U}_q(u(r))$ in the $q$-Fock space is constructed. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-05T10:36:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "20G42",
      "20G45",
      "81Q12",
      "14M12",
      "16T10",
      "81R50",
      "81Q05",
      "81Q10"
    ],
    "keywords": [
      "canonical commutation relations",
      "negative non-compact root spaces",
      "determinantal ideals",
      "second order creation operators",
      "fock space"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}