Analysis of the solution space for the quantum KZ equation and its applications to integrable systems

得到了量子Knizhnik-Zamolodchikov方程相关问题的两个结果。首先，我们构造了一个代数，它描述了包含多个zeta值的q-类似物的q-级数族的乘法结构。在我们的框架中，构造并证明了一类称为双洗牌关系的线性关系。其次，我们定义了具有δ势的非理想玻色气体的哈密顿量的离散模拟，并利用仿射Hecke代数的表示，通过Bethe ansatz方法构造了本征函数。

We get two results about related problems to the quantum Knizhnik-Zamolodchikov equation. First, we constructed an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. A family of linear relations called the double shuffle relations is formulated and proved in our framework. Second, we defined a discrete analogue of the Hamiltonian of the non-ideal Bose gas with delta-potentials, and constructed eigenfunctions by means of the Bethe ansatz method making use of a representation of the affine Hecke algebra.