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.

两年来取得了两项主要成就。第一个是关于费米子公式和相关的话题。我们将费米子公式推广到所有的扭量子仿射代数，并对路径实现中的张量积定理、旋子特征公式、双对数和规则、Kirillov-Reshetikhin猜想、Bethe代数在q = 1和q = 0时的完备性等方面给出了统一的描述.第二个是关于由晶体基构造的孤立子元胞自动机。我们证明了对于小秩代数，孤立子的散射规则与组合规则R是一致的。我们证明了在无限载体的情况下，时间演化被分解为Weyl群算子的乘积。这大大简化了对粒子和反粒子运动的动力学描述，这些粒子和反粒子经历了对的产生和湮灭。对于An^<（1）>情形，我们通过探索与离散KP方程的联系构造了N-孤子解.

项目成果

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)>

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 (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）。

G. Hatayama, et al: "Scattering rules in soliton cellular automata associated with crystal bases"Contemporary Math. (AMS). (in press).

G. Hatayama 等人：“与晶体基相关的孤子元胞自动机中的散射规则”当代数学。