Hall algebras revisited

Ringel CM. Hall algebras revisited. Preprint / Universität Bielefeld, Sonderforschungsbereich 343, Diskrete Strukturen in der Mathematik ; 92,29. Bielefeld; 1992.For any finite-dimensional algebra over a finite field, the corresponding Hall algebra has been introduced in order to handle the possible filtrations of modules with fixed factors. For the path algebra of a Dynkin diagram with a fixed orientation, it has been shown that the Hall algebra satisfies relations which are similar to the Drinfeld-Jimbo relations defining quantum groups, but they depend on the chosen orientation. The purpose of this note is to adjust the multiplication of a Hall algebra in order to obtain the Drinfeld-Jimbo relations themselves. The additional factor introduced in our change of multiplication involves the Euler characteristic, in this way we eliminate the dependence on the orientation