According to the canonical isomorphism between the positive part U+ q (g) of the Drinfeld–Jimbo quantum group Uq(g) and the generic composition algebra C(¢) of ¤, where the Kac–Moody Lie algebra g and the finite dimensional hereditary al- gebra ¤ 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 com- mutation relations among all root vectors are given and an integral PBW–basis of this algebra is also obtained.