Perfect crystals andq-deformed Fock spaces

Abstract

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.