quantum groups and double quiver algebras

Let Uq(g) be the Drinfeld-Jimbo quantum group, which is a deformation of the universal enveloping algebra of a finite dimensional semisimple Lie algebra g. In the generic case, i.e. the parameter q is not a root of unity, several models have been raised to realize it. For example, the Ringel-Hall algebra approach is one successful model among them, see [10, 5, 11]. The case where q is a root of unity is of an particular interest since it is related with the modular representation theory. It is remarkable that a finite dimensional Hopf algebra, so-called restricted version of Uq(g) arose naturally when Lusztig considered this class of quantum The author is partially supported by the National Science Foundation of China (Grant. 10271014) and Natural Science Foundation of Beijing City(Grant. 1042001) 2000 Mathematics Subject Classification. 81R50, 16W30.