Root vectors of the composition algebra of the Kronecker algebra

Abstract

According to the canonical isomorphism between the positive part \({\bf U}^+_q({\bf g})\) of the Drinfeld-Jimbo quantum group \({\bf U} _q ({\bf g})\) and the generic composition algebra \({\mathcal C} (\Delta)\) of \(\Lambda\), where the Kac-Moody Lie algebra \({\bf g}\) 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.