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.

与量子基团和相关主题相关的一维孤子细胞自动机已经取得了重大进展。在此解释了三年中获得的结果（1） - （6）。（1）在D型热带R上，发现其双线性形式，并且显示了DKP HierArch的Tau功能，显示了DKP HierArch的TAU功能。通过系统的减少，获得了C型类型和扭曲的仿射谎言代数的相似结果。原始Q = 0。引入了两种规范，并研究了其财产。（4）构建了具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。在热带环境中发现了边界的整合性条件的解决方案。（5）对KKR Belive的新描述，KKR Belive，证明效率公式的CRUX在Crytal Base理论中以类似的形式在Crytal Base理论中纯粹地获得了纯粹的Crypinal r cy。结果使盒球系统的反向散射形式主义大声。（6）周期性的盒球系统扩展到kkm晶体，并在状态计数公式和通用动力学时期内提出了km晶体的晶体案例和猜想。对于符号限制了一个类型的情况，初始值问题是通过统一的bethe bethe Ansat ansatz ansat an ansats and q = 0.0 = 0 = 0 = 0 = 0 = 0.