Perfect Crystals and q-deformed Fock Spaces

Abstract

A general scheme for the wedge construction of q-deformed Fock spaces using the theory of perfect crystals is presented. Let $U_q(\g)$ be a quantum affine algebra. Let $V$ be a finite-dimensional $U'_q(\g)$-module with a perfect crystal base of level~$l$. Let $V_\aff\simeq V\otimes\C[z,z^{-1}]$ be the affinization of $V$, with crystal base $(L_\aff,B_\aff)$. The wedge space $V_\aff\wedge V_\aff$ is defined as the quotient of $V_\aff\otimes V_\aff$ by the subspace generated by the action of $U_q(\g)[z^a\otimes z^b +z^b\otimes z^a]_{a,b\in\Z}$ on $v\otimes v$ ($v$ an extremal vector). The wedge space $\bigwedge^r V_\aff$ ($r\in\N$) is defined similarly. Normally ordered wedges are defined by using the energy function $H:B_\aff\otimes B_\aff\to\Z$. Under certain assumptions, it is proved that normally ordered wedges form a base of $\bigwedge^r V_\aff$. A q-deformed Fock space is defined as the inductive limit of $\bigwedge^r V_\aff$ as $r\to\infty$, 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 $U_q(\g)$-module. An action of the bosons, which commute with the $U'_q(\g)$-action, is given on the Fock space. It induces the decomposition of the q-deformed Fock space into the tensor product of an irreducible $U_q(\g)$-module and a bosonic Fock space. As examples, Fock spaces for types $A^{(2)}_{2n}$, $B^{(1)}_n$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$ and $D^{(2)}_{n+1}$ at level~1 and $A^{(1)}_1$ 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.