Comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver

The aim of this paper is to construct comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver, where $q$ is not a root of unity. By embedding the quantized algebra $U_q(sl_2)$ into the path coalgebra $k\mathcal{D}^c$, where $\mathcal{D}$ is the Gabriel quiver of $U_q(sl_2)$ as a coalgebra, we obtain a basis of $U_q(sl_2)$ in terms of combinations of paths in the quiver $\mathcal{D}$; this special basis enable us to describe the category of $U_q(sl_2)$-comodules by certain representations of $\mathcal{D}$; and this description further permits us to construct a class of modules of the quantum special linear group $SL_q(2)$, from certain representations of $\mathcal{D}$, via the duality between $U_q(sl_2)$ and $SL_q(2)$.