A realization of quantum groups via product valued quivers

Let C be the symmetrizable generalized Cartan matrix associated to a valued quiver Γ→. We show that the whole quantum group associated to C can be realized from the abelian category of the representations of the product valued quiver Γ→^± of Γ→ with the Kronecker quiver. This method can be used to realize the whole generalized Kac-Moody Lie algebra associated to C, as discussed in Li and Lin (submitted for publication, arXiv:math/0610448).