Drinfeld Double and Ringel–Green Theory of Hall Algebras

Abstract

As a continuation of the work of Ringel and Green on Hall algebras, the Hopf algebra structure of a (extended twisted) Hall algebra is presented here. By introducing the Ringel pairing of Hall algebras, the Drinfeld double of a Hall algebra is constructed. The reduced form of this Drinfeld double is applicable to the generic composition algebra, which gives a complete realization of quantum group. This Ringel–Green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of Hall algebras.