A Realization of Quantum Groups via Product Valued Quivers

Let C be the symmetrizable generalized Cartan matrix associated to a valued quiver \(\overset{\rightarrow}{\Gamma}\). 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 \(\overset{\rightarrow}{\Gamma^{\pm}}\) of \(\overset{\rightarrow}{\Gamma}\) 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).