Crystal basis in quantum groups and its applications

两年来主要有两个成果。第一个是关于费米子公式和相关主题。我们将费米子公式推广到所有扭曲量子仿射代数，并对路径实现中的张量积定理、自旋特征公式、双对数和规则、Kirillov-Reshetikhin猜想、q=1和q=0时Bethe ansatz的完备性等各个方面给出了统一的描述。所有非例外代数的组合R也是通过显式插入算法获得的。第二个是关于从晶体基构造的孤子元胞自动机。我们证明了孤子的散射规则与较小阶代数的组合 R 相同。我们在无限载波情况下证明了时间演化被分解为 Weyl 群算子的乘积。这极大地简化了对粒子和反粒子运动的动力学描述，这些运动经历了粒子对的产生和湮灭。对于 An^<(1)> 情况，我们通过探索与离散 KP 方程的联系来构造 N-孤子解。

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.