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Perfect crystals and \(q\)-deformed Fock spaces. (English) Zbl 0959.17014
Summary: In [E. Stern, Int. Math. Res. Not. 4, 201-220 (1995; Zbl 0823.17042) and M. Kashiwara, T. Miwa and E. Stern, Sel. Math. 1, 787-805 (1995; Zbl 0857.17013)] the semi-infinite wedge construction of level 1 \(U_q(A_n^{(1)})\) Fock spaces and their decomposition into the tensor product of an irreducible \(U_q(A_n^{(1)})\)-module and a bosonic Fock space are given. Here a general scheme for the wedge construction of \(q\)-deformed Fock spaces using the theory of perfect crystals is presented.
Let \(U_q({\mathfrak g})\) be a quantum affine algebra. Let \(V\) be a finite-dimensional \(U_q'({\mathfrak g})\)-module with a perfect crystal base of level \(l\). Let \(V_{\text{aff}} \simeq V\otimes \mathbb{C} [z,z^{-1}]\) be the affinization of \(V\), with crystal base \((L_{\text{aff}}, B_{\text{aff}})\). The wedge space \(V_{\text{aff}} \wedge V_{\text{aff}}\) is defined as the quotient of \(V_{\text{aff}} \otimes V_{\text{aff}}\) by the subspace generated by the action of \(U_q ({\mathfrak g}) [z^a\otimes z^b+ z^b \otimes z^a]_{a,b\in \mathbb{Z}}\) on \(v\otimes v\) (\(v\) an extremal vector). The wedge space \(\bigwedge^r V_{\text{aff}}\) \((r\in \mathbb{N})\) is defined similarly. Normally ordered wedges are defined by using the energy function \(H: B_{\text{aff}} \otimes B_{\text{aff}}\to \mathbb{Z}\). Under certain assumptions, it is proved that normally ordered wedges form a base of \(\bigwedge^r V_{\text{aff}}\).
A \(q\)-deformed Fock space space is defined as the inductive limit of \(\bigwedge^r V_{\text{aff}}\) as \(r\to \infty\), taking 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 of Fock space has the structure of an integrable \(U_q({\mathfrak g})\)-module. An action of the bosons, which commute with the \(U_q'({\mathfrak g})\)-action, is given on the Fock space. It induces the decomposition of the \(q\)-deformed Fock space into the tensor product of an irreducible \(U_q({\mathfrak g})\)-module and a bosonic Fock space.
As examples, Fock spaces for types \(A_{2n}^{(2)}\), \(B_n^{(1)}\), \(A_{2n-1}^{(2)}\), \(D_n^{(1)}\) and \(D_{n+1}^{(2)}\) 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.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B69 Vertex operators; vertex operator algebras and related structures
Full Text: DOI
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