Quantum groups and discrete integrable system

与量子群相关的一维孤立子元胞自动机及其相关课题的研究取得了重要进展，本文解释了三年来所得到的结果（1）-（6）。(1)在D型热带李代数R上，找到了它的双线性型，并证明了DKP族的τ函数是它的解.通过系统约化，对C型仿射李代数和扭A也得到了类似的结果. (2)对于容量大于1的元胞自动机，时间演化用粒子和反粒子的运动进行显式算法描述，粒子和反粒子经历对的产生和湮灭. (3)从顶点模型的某个极限出发，构造了盒球系统的量子化，该量子化在q=0时趋向于原量子化。引入了两类范数，并研究了它们的性质。(4)构造了一个具有反射端的箱球系统。提取了孤立子的自由度，阐明了散射和反射规律，并在热带环境下求解了边界可积条件。(5)利用晶体基理论中的组合R，对费米子公式证明中的关键问题KKR双射给出了一种新的描述，并对KKM晶体的其他情况给出了类似的描述.得到了箱球系统的逆散射公式。(6)将周期箱球系统推广到KKM晶体和A型KM晶体情形，给出了系统的状态计数公式和动力学周期的表达式，并对最辛A型情形，通过统一q=0和q=1时的Bethe估计，完全解决了系统的初值问题.

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.