Title: About 30 Years of Integrable Chiral Potts Model, Quantum Groups at Roots of Unity and Cyclic Hypergeometric Functions Show Chinese title

Title: 关于可积 chiral Potts 模型的 30 年、根单位的量子群与循环超几何函数

Abstract: In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang--Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts model leads to special identities for basic hypergeometric series in the $q$ a root-of-unity limit. Many of the well-known summation formulae for basic hypergeometric series do not work in this case. However, if we require the summand to be periodic, then there are many summable series. For example, the integrability condition, namely, the star-triangle equation, is a summation formula for a well-balanced ${}_4\Phi_3$ series. We finish with a few remarks about the relation with quantum groups at roots of unity. Show Chinese abstract

Abstract: 本文我们讨论可积的 chiral Potts 模型，因为它清楚地关系到我们如何结识了沃恩·琼斯，我们在秦皇岛会议上庆祝了他的生日。 令人惊讶的是，我们还可以庆祝这个模型的生日，因为大约 30 年前它作为第一个用高亏格函数参数化的星-三角方程解被引入。 在介绍最一般的棋盘式 Yang-Baxter 方程后，我们将专注于星-三角方程，并讨论它与纽结理论的关系。 然后我们展示可积的 chiral Potts 模型如何在 $q$ 单位根极限下导出基本超几何级数的一些特殊恒等式。 许多已知的基本超几何级数求和公式在这种情况下不起作用。 然而，如果我们要求和项具有周期性，则有许多可求和的级数。 例如，可积性条件，即星-三角方程，是一个关于平衡良好的 ${}_4\Phi_3$ 级数的求和公式。 最后，我们简单讨论了它与单位根量子群的关系。