Perfect crystals andq-deformed Fock spaces

In [S], [KMS] the semi-infinite wedge construction of level 1U _q (A _n ⁽¹⁾ ) Fock spaces and their decomposition into the tensor product of an irreducibleU _q (A _n ⁽¹⁾ )-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 [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,b onv v (v an extremal vector). The wedge space ^r V _aff (r ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :B _aff B _aff . 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 _2n ⁽²⁾ ,B _n ⁽¹⁾ ,A _2n ^–1/(2) ,D _n ⁽¹⁾ andD _n ^+1/(2) 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.