Quantum groups and discrete integrable system

一维孤子元胞自动机在与量子群及相关问题相关的研究中取得了重大进展.本文解释了三年来的结果(1)-(6).在D型热带R上，找到了它的双线性形式，并证明了DKP方程的tau函数是它的解.(2)对于容量大于1的元胞自动机，用显式算法描述了粒子和反粒子的运动，并对粒子和反粒子的运动进行了成对和湮灭。(3)从顶点模型的某一极限出发，构造了盒球系统的量子化，该量子化在Q=0时趋于原始的量子化。引入了两种范数，并研究了它们的性质。(4)构造了一个带反射端的箱球系统。提取了孤子的自由度，阐明了散射和反射规则。在热带背景下找到了边界可积性条件的解。(5)利用结晶基理论中的组合R，得到了证明费米子公式的关键--KKR双射的新描述。对所有其他KKM晶体的情况也给出了类似的描述。将周期箱球系统推广到KKM晶体和A型KM晶体情况，提出了关于状态计数公式和一般动力学周期的猜想；对于最复杂的A型情况，通过统一Q=0和Q=1的Bethe ansatz，完全解决了初值问题。

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.