Crystal basis in quantum groups and its applications

量子群中的晶体基础及其应用

• 批准号: 12640385
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640385/
• 关键词: statistical mechanics; solvable lattice models; Yang-Baxter equation; quantum groups; affine Lie algebra; crystal basis; soliton cellular automaton; Bethe ansatz; ソリトンセルオートマトン; セン・バクスター方程式

There are two main achievements in the two years. The first one is about the fermionic formula and related topics. We generalized the fermionic formula to all the twisted quantum affine algebras and gave a unified description on various aspects including the tensor product theorem in path realization, spinon character formulae, dilogarith sum rules, the Kirillov-Reshetikhin conjecture, completeness of Bethe ansatz both at q = 1 and q = 0 and so forth. Combinatorial R for all non exceptional algebras is also obtained in terms of an explicit insertion algorithm.The second one is about the soliton cellular automata constructed from the crystal basis. We proved that the scattering rule of solitions is identical with the combinatoral R for smaller rank algebra. We proved in the infinite carrier case that the time evolution is factorized into a product of Weyl group operators. This greatly simplified the description of the dynamics into the motion of particles and antiparticles that undergo pair creation and annihilation. For An^<(1)> case, we constructed N-soliton solution by exploring the connection to the discrete KP equation.

在这两年中，有两个主要成就。第一个是关于费米的公式和相关主题。我们将费米金公式概括为所有扭曲的量子仿射代数，并就路径实现的张量定理，Spinon角色公式，Dilogarith Sum规则，Kirillov-Reshetikhin猜想，Bethe Ansatz在Q = 1和Q = 0和Quort quth of qut = 0和等等。也根据明确的插入算法获得了所有非特殊代数的组合R，第二个是关于从晶体基础构建的孤子细胞自动机。我们证明，对于较小等级代数的组合R的散射规则与组合R相同。我们在无限的载体案例中证明，时间演化被分解为Weyl组运营商的产物。这大大简化了动力学的描述为经历成对创造和an灭的颗粒和反粒子的运动。对于An^<（1）>情况，我们通过探索与离散KP方程的连接来构建N-Soliton解决方案。

项目成果

K.Fukuda, M.Okado, Y.Yamada: "Energy functions in box ball systems"Int. J. Mod. Phys. A. 15. 1379-1392 (2000)

K.Fukuda、M.Okado、Y.Yamada：“盒子球系统中的能量函数”Int。

G.Hatayama, A.Kuniba, M.Okado, T.Takagi: "Combinatorial R matrices for a family of crystals : B^<(1)>_n, D^<(1)>_n A^<(2)>_<2n> and D^<(2)>_<n+1> cases"J. Alg.. 247. 577-615

G.Hatayama、A.Kuniba、M.Okado、T.Takagi：“晶体族的组合 R 矩阵：B^<(1)>_n、D^<(1)>_n A^<(2)>

G.Hatayama, A.Kuniba, M.Okado, T.Takagi: "Combinatorial R matrices for a family of crystals : B^<(1)>_n, D^<(1)>_n, A^<(2)>_<2n> and D^<(2)>_<n+1> cases"J. Alg.. 247. 577-615 (2002)

G.Hatayama、A.Kuniba、M.Okado、T.Takagi：“晶体族的组合 R 矩阵：B^<(1)>_n、D^<(1)>_n、A^<(2)

A.Kuniba, T.Nakanishi, Z.Tsuboi: "The Bethe equation at q=0, the Mobius inversion formula, and weight multiplicities : III. The X^<(r)>_N case"Lett. Math. Phys.. (掲載予定).

A.Kuniba、T.Nakanishi、Z.Tsuboi：“q=0 时的 Bethe 方程、莫比乌斯反演公式和权重重数：III. X^<(r)>_N 情况”Lett。 ..（预定出版）。

A. Kuniba, et al: "Difference L operators related to q-characters"J. Phys. A. Math. and Gen. (in press).

A. Kuniba 等人：“与 q 字符相关的差分 L 运算符”J。