Representation theory of elliptic quantum groups and its application

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1. Formulation of the ABF model in regime II in terms of the vertex operatorsSupposing the local height restriction is κ + 1, we have identified the half-infinite row and column transfer matrices and the creation operator of the physical excitation with the vertex operators and the basic q-parafermion currents of the q-deformed Ζ_κ parafermion theory, respectively. We have also shown that the scaling limit of the model is given by Koberle-Swieca's factorized scattering theory.2. Representation of <sl>^^^^_2 in the principal gradationWe have given a free field realization of the representation of <sl>^^^^_2 in the generic level principal gradation. Especially, we have given a free field realization of Lepowsky-Wilson's Ζ-algebra as the deformed Virasoro algebra Vir_<q, t> in the limit (t = -q^<(k+22)/2>, q → 1).3. Higher level representation of the elliptic quantum group β_<q,【lambda bar】>(<sl>^^^^_2)Based on the free field realization of β_<q,【lambda bar】>(<sl>^^^^_2) and the elliptic algebra U_<q, p>(<sl>^^^^_2), we have given a systematic construction of the higher spin intertwining operators of β_<q,【lambda bar】>(<sl>^^^^_2). We have also clarified a relation between the coset decomposition of the tensor product of two U_q(<sl>^^^^_2)-modules and the representation of U_<q, p>(<sl>^^^^_2).