Quantum groups and discrete integrable system

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Significant progress has been made on the one dimensional soliton cellular automata associated with quantum groups and related topics.The results (1)-(6) obtained in the three years are explained herewith.(1) On D type tropical R, its bilinear form was found and the tau functions of the DKP hierarchy was shown to be a solution of it. By a systematic reduction, similar results were obtained for the affine Lie algebras of type C type and twisted A.(2) For the cellular automata with capacity greater than one, the time evolution was described with an explicit algorithm in terms of the motion of particles and antiparticles which undergo the pair creation and annihilation.(3) A quantization of the box-ball system was constructed from a certain limit of a vertex model, which tends to the original one at q=0. Two kinds of norm were introduced and their property was investigated.(4) A box-ball system with a reflecting end was constructed. The soliton degrees of freedom was extracted, scattering and reflection rules are clarified.A solution of the boundary integrability condition is found at the tropical setting.(5) A new description of the KKR bijection, the crux in proving the fermionic formula, was obtained purely in terms of the combinatorial R in crytal base theory.A similar description was conjectured for all the other KKM crystal case. The result yiled the inverse scattering formalism of the box-ball system.(6) Periodic box-ball systems were extended to the KKM crystal and A type KM crystal cases and conjectures were put forward on the state counting formula and the generic dynamical period.For the symplest A type case, the initial value problem was completely solved by unifying the Bethe ansatz at q=0 and q=1.