Study of integrable structure inquanteem integrable model

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We obtain the following new results on the integrable structure of integrable lattice models.1. The formal completeness theorem on the Bethe equation for the XXZ-type spin models. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))By classifying the string solutions of the Bethe equation for the XXZ-type spin models in the q=0 limit, we showed that, under the Kirillov-Reshetikhin (KR) conjecture on the KR modules of the quantum affine algebras, the power series formula of the character of the KR module representing the formal completeness of the XXZ-type spin models holds for any affine Lie algebra.2. Reformulation of the Kirillov-Reshetikhin conjecture by the canonical solutions of the Q-systems. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.))By observing the principle mechanism for various formulae and theorems obtained in the study of 1, we completely clarified the relation between these power series formulae representing the formal completeness and the underlying functional (algebraic) equations (Q-system of KR-type). Namely, we introduce a kind of functional equations (finite Q-system) for a finite number of functions of a finite number of variables. A finite Q-system has a remarkable property that its unique solution admits two power series formula by the Lagrange inversion formula for several variables. They are exactly a finite-variable analogue of the formal completeness formulae for the XXX-type and XXZ-type models. The formal completeness formula can be obtained as the projective limit of this formula.