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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.

##### MSC:
 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
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