Perfect Crystals and q-deformed Fock Spaces

A b s t r a c t . In [S], [KMS] the semi-infinite wedge construction of level 1 Uq(A (1)) Fock spaces and their decomposition into the tensor product of an irreducible Uq(A(1))-module and a bosonic Fock space were given. Here a general scheme for the wedge construction of q-deformed Fock spaces using the theory of perfect crystals is presented. Let Uq(fJ) be a quantum affine algebra. Let V be a finite-dimensional U£(g)-module with a perfect crystal base of level t. Let Vaf~ ~ V ® C[z, z -1] be the affinization of V, with crystal base (Laff, Baff). The wedge space Vaff A Vag is defined as the quotient of V~ff ® Vag by the subspace generated by the action of Uq(g)[z ~ ® z b + z b ® z~]a,beZ on v ® v (v an extremal vector). The wedge space A T Vaff (r E N) is defined similarly. Normally ordered wedges are defined by using the energy function H : /?aft ® Baff --~ Z. Under certain assumptions, it is proved that normally ordered wedges form a base of A ~ ~f f . A q-deformed Fock space is defined as the inductive limit of A r Vaff as r --~ o0, 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 integrable Uq(9)-module. An action of the bosons, which commute with the U£(9)-action, is given on the Fock space. It induces the decomposition of the q-deformed Fock space into the tensor product of an irreducible Uq(g)-modute and a bosonic Fock space. As examples, Fock spaces for types a(2) B~ 1), a(2) D O) and n(2) "'2n~ "~2n-l' ~n-bl at level 1 and A~ 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.