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

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