Root vectors of the composition algebra of the Kronecker algebra

According to the canonical isomorphism between the positive part $U^+_q(g)$ of the Drinfeld–Jimbo quantum group $U_q (g)$ and the generic composition algebra ${\mathcal C} (\Delta)$ of $\Lambda$, where the Kac–Moody Lie algebra $g$ and the finite dimensional hereditary algebra $\Lambda$ have the same diagram, in specially, we get a realization of quantum root vectors of the generic composition algebra of the Kronecker algebra by using the Ringel–Hall approach. The commutation relations among all root vectors are given and an integral PBW–basis of this algebra is also obtained.