{
  "annotation_id": "907bb893-ba9a-446a-bd69-52baf0d9ea44",
  "date_created": "2026-03-02T18:01:28.135000Z",
  "date_modified": "2026-03-02T18:01:28.135000Z",
  "file_hash": "8046c36cdca156b68b90e07a88b5a7ff9492eb6d81c6b4e61514fe2cd039a0d4",
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  "record": {
    "abstract": "A general scheme for the wedge construction of q-deformed Fock spaces using\nthe theory of perfect crystals is presented.\n  Let $U_q(\\g)$ be a quantum affine algebra. Let $V$ be a finite-dimensional\n$U\u0027_q(\\g)$-module with a perfect crystal base of level~$l$. Let $V_\\aff\\simeq\nV\\otimes\\C[z,z^{-1}]$ be the affinization of $V$, with crystal base\n$(L_\\aff,B_\\aff)$. The wedge space $V_\\aff\\wedge V_\\aff$ is defined as the\nquotient of $V_\\aff\\otimes V_\\aff$ by the subspace generated by the action of\n$U_q(\\g)[z^a\\otimes z^b +z^b\\otimes z^a]_{a,b\\in\\Z}$ on $v\\otimes v$ ($v$ an\nextremal vector). The wedge space $\\bigwedge^r V_\\aff$ ($r\\in\\N$) is defined\nsimilarly. Normally ordered wedges are defined by using the energy function\n$H:B_\\aff\\otimes B_\\aff\\to\\Z$. Under certain assumptions, it is proved that\nnormally ordered wedges form a base of $\\bigwedge^r V_\\aff$.\n  A q-deformed Fock space is defined as the inductive limit of $\\bigwedge^r\nV_\\aff$ as $r\\to\\infty$, taken along the semi-infinite wedge associated to a\nground state sequence. It is proved that normally ordered wedges form a base of\nthe Fock space and that the Fock space has the structure of an integrable\n$U_q(\\g)$-module. An action of the bosons, which commute with the\n$U\u0027_q(\\g)$-action, is given on the Fock space. It induces the decomposition of\nthe q-deformed Fock space into the tensor product of an irreducible\n$U_q(\\g)$-module and a bosonic Fock space.\n  As examples, Fock spaces for types $A^{(2)}_{2n}$, $B^{(1)}_n$,\n$A^{(2)}_{2n-1}$, $D^{(1)}_n$ and $D^{(2)}_{n+1}$ at level~1 and $A^{(1)}_1$ at\nlevel~$k$ are constructed. The commutation relations of the bosons in each of\nthese cases are calculated, using two point functions of vertex operators.",
    "arxiv_id": "q-alg/9603025",
    "authors": [
      "Masaki Kashiwara",
      "Tetsuji Miwa",
      "Jens-Ulrik H. Petersen",
      "Chong Ming Yung"
    ],
    "categories": [
      "q-alg",
      "hep-th",
      "math.QA"
    ],
    "journal_ref": "Selecta Mathematica 2 (1996) no.3 415-499",
    "title": "Perfect Crystals and q-deformed Fock Spaces",
    "url": "https://arxiv.org/abs/q-alg/9603025"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "81edec97-b0b9-47da-bd4e-3cde5f2f7644",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}