A是有限域k上的有限维遗传代数，HRv(A)与CRv(〓)分别是 Ringel-Hall代数与Composition代数。通过引入HRV(A)中的左(右)导子 αδ(δα)αεP，使得〓的Exceptional序列及Braid群对 Exceptional 序列的作用，提供了计算根向量的有效算法，即我们给出了把{Uλ｜λεP，Uλ是Exceptional模}表达成{U｝ ieI 的组合表达式的精确递推公式。我们计算的基础是具有两个单模同构类的有限域上的有限维遗传代数。所得的结果不仅适合于半单李代数的量子包络代数，而且还适合于一切可对称化的Kac-Moody代数的量子包络代数。

Let A be a finite dimensional hereditary algebra over a finite field k,HRv(〓)and CRV(〓)be respectively the Ringel-Hall algebra and the composition algebra of A.Define αδ(δα),where αεP,to be the left (right) derivation of Hav(A). By using the exceptional sequence of A-mod and the action of the braid group on the exceptional sequences, we provide an effective algorithm of calculating root vectors.We can express {Uλ|λεP,Uλ is an exceptional A-mod} as the combination of {Uj| iεI}.The finite dimensional hereditary algebra over a finite field with two isomorphism classes of simple modules is basis of our calculation. In general, our result is not only for the quantum enveloping algebrathe semi-simple Lie algebra, but also for the quantum enveloping algebra of the symmetrizable Kac-Moody algebra.