Resumen de Perfect crystals andq-deformed Fock spaces

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

• 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. LetVaff ≏V ⊗ ℂ[z,z−1] be the affinization ofV, with crystal base (Laff,Baff). The wedge spaceVaff ∧Vaff is defined as the quotient ofVaff ⊗Vaff 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 ∧rVaff (r ε ℕ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :Baff ⊗Baff → ℤ. Under certain assumptions, it is proved that normally ordered wedges form a base of ∧rVaff.

  Aq-deformed Fock space is defined as the inductive limit of ∧rVaff 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.