For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.
A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form.Their positive parts turn out to be 2-cocycle deformations of each other under some conditions.An operator realization of the positive part is given.
HU NaiHong PEI YuFeng Department of Mathematics,East China Normal University,Shanghai 200062,China
