dorsal/arxiv
View SchemaPerfect Crystals and q-deformed Fock Spaces
| Authors | Masaki Kashiwara, Tetsuji Miwa, Jens-Ulrik H. Petersen, Chong Ming Yung |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9603025 |
| URL | https://arxiv.org/abs/q-alg/9603025 |
| Journal | Selecta Mathematica 2 (1996) no.3 415-499 |
Abstract
A general scheme for the wedge construction of q-deformed Fock spaces using the theory of perfect crystals is presented. Let $U_q(\g)$ be a quantum affine algebra. Let $V$ be a finite-dimensional $U'_q(\g)$-module with a perfect crystal base of level~$l$. Let $V_\aff\simeq V\otimes\C[z,z^{-1}]$ be the affinization of $V$, with crystal base $(L_\aff,B_\aff)$. The wedge space $V_\aff\wedge V_\aff$ is defined as the quotient of $V_\aff\otimes V_\aff$ by the subspace generated by the action of $U_q(\g)[z^a\otimes z^b +z^b\otimes z^a]_{a,b\in\Z}$ on $v\otimes v$ ($v$ an extremal vector). The wedge space $\bigwedge^r V_\aff$ ($r\in\N$) is defined similarly. Normally ordered wedges are defined by using the energy function $H:B_\aff\otimes B_\aff\to\Z$. Under certain assumptions, it is proved that normally ordered wedges form a base of $\bigwedge^r V_\aff$. A q-deformed Fock space is defined as the inductive limit of $\bigwedge^r V_\aff$ as $r\to\infty$, taken along the semi-infinite wedge associated to a ground state sequence. It is proved that normally ordered wedges form a base of the Fock space and that the Fock space has the structure of an integrable $U_q(\g)$-module. An action of the bosons, which commute with the $U'_q(\g)$-action, is given on the Fock space. It induces the decomposition of the q-deformed Fock space into the tensor product of an irreducible $U_q(\g)$-module and a bosonic Fock space. As examples, Fock spaces for types $A^{(2)}_{2n}$, $B^{(1)}_n$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$ and $D^{(2)}_{n+1}$ at level~1 and $A^{(1)}_1$ at level~$k$ are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators.
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"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"
},
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