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Quantum groups and discrete integrable system

量子群和离散可积系统

基本信息

批准号：
15540363
负责人：
KUNIBA Atsuo
金额：
$ 1.22万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540363/
关键词：
integrable system solvable lattice model Yang-Baxter equation Bethe ansatz quantum group crystal base ultradiscretization box-ball system ソリトンセルオートマトン

项目摘要

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.

在与量子群相关的一维孤子元胞自动机及相关课题方面取得了重大进展。现对三年来得到的结果(1)-(6)进行解释。(1)在D型热带R上，找到了它的双线性形式，并证明了DKP层次的tau函数是它的解。通过系统约简，对于C型仿射李代数和扭曲A型仿射李代数也得到了类似的结果。(2)对于容量大于1的元胞自动机，用显式算法用粒子和反粒子的运动来描述时间演化，粒子和反粒子经历粒子对的产生和湮灭。(3)从顶点模型的一定极限出发构造了盒球系统的量化，该系统趋向于原始的q=0。介绍了两种范数并研究了它们的性质。(4)构造了带有反射端的盒球系统。提取了孤子自由度，阐明了散射和反射规则。找到了热带环境下边界可积条件的解。(5)纯粹根据晶基理论中的组合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晶例也做出了类似的描述。 (6)将周期盒球系统推广到KKM晶体和A型KM晶体情况，并提出了状态计数公式和一般动态周期的猜想。对于最简单的A型情况，通过统一q=0和q=1时的Bethe ansatz完全解决了初值问题。