Comodules of Uq(sl2) and modules of SLq(2) via quiver methods

The aim of this paper is to construct comodules of Uq(sl2) and modules of SLq(2) via quiver, where q is not a root of unity. By embedding Uq(sl2) into the path coalgebra image, where image is the Gabriel quiver of Uq(sl2) as a coalgebra, we obtain a basis of Uq(sl2) in terms of combinations of paths of the quiver image; this special basis enables us to describe the category of Uq(sl2)-comodules by certain representations of image; and this description further permits us to construct a class of modules of SLq(2), from certain representations of image, via the duality between Uq(sl2) and SLq(2).