Tensor product modules of $U_q(\mathfrak{sl}_2)$ and Clebsch-Gordan type formulas

In this talk, we will construct and classify rank one $\mathbb{C}[K^{\pm1}]$-free modules for the quantum group $U_q(\mathfrak{sl}_2)$. Then we study their tensor products with finite-dimensional simple modules. We obtain some direct sum decomposition formulas for the tensor products, which are similar to the classical Clebsch-Gordan formulas. This talk is based on joint works with Yan-an Cai, Xiangqian Guo, Yao Ma and Mianmian Zhu.