Perfect crystals andq-deformed Fock spaces

In [S], [KMS] the semi-infinite wedge construction of level 1U_q (A ⁽¹⁾_n ) Fock spaces and their decomposition into the tensor product of an irreducibleU_q (A ⁽¹⁾_n )-module and a bosonic Fock space were given. Here a general scheme for the wedge construction ofq-deformed Fock spaces using the theory of perfect crystals is presented.

LetU_q(g) be a quantum affine algebra. LetV be a finite-dimensionalU′_q (g)-module with a perfect crystal base of levell. LetV_aff ≏V ⊗ ℂ[z,z⁻¹] be the affinization ofV, with crystal base (L_aff,B_aff). The wedge spaceV_aff ∧V_aff is defined as the quotient ofV_aff ⊗V_aff by the subspace generated by the action ofU_q (g) [z^a ⊗z ^b+z^b ⊗z^a]_a,bεℤ onv ⊗v (v an extremal vector). The wedge space ∧^r V_aff (r ε ℕ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :B_aff ⊗B_aff → ℤ. Under certain assumptions, it is proved that normally ordered wedges form a base of ∧^r V_aff.

Aq-deformed Fock space is defined as the inductive limit of ∧^r V_aff asr → ∞, 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 integrableU_q(g)-module. An action of the bosons, which commute with theU′_q(g)-action, is given on the Fock space. It induces the decomposition of theq-deformed Fock space into the tensor product of an irreducibleU_q(g)-module and a bosonic Fock space.

As examples, Fock spaces for typesA ⁽²⁾_2n ,B ⁽¹⁾_n ,A ^−1/(2)_2n ,D ⁽¹⁾_n andD ^+1/(2)_n at level 1 andA ⁽¹⁾₁ at levelk are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators.

Authors and Affiliations

1. Research Institute for Mathematical Sciences, Kyoto University, 606-01, Kyoto, Japan

   M. Kashiwara, T. Miwa, J. -U. H. Petersen & C. M. Yung

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