Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a nite dimensional hereditary algebra over a nite eld k; H(() and C(() be respectively the Hall algebra and the composition algebra of ; P be the isomorphism classes of nite dimensional-modules and I the isomorphism classes of simple-modules. We deene and , 2 P, to be the right and left derivations of H(() respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of-mod , we provide an eeective algorithm of calculating root vectors. This means that we get an inductive method to express u as the combinations of elements u i in the Hall algebra, where i 2 I and 2 P is any exceptional-module. Because of the canonical isomorphism between the Drinfeld-Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups.