The Hall algebra approach to Drinfeld's presentation of quantum loop algebras

The quantum loop algebra U-v(Lg) was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra g. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra U-v(Lg), for some g with a star-shaped Dynkin diagram. In this paper we study Drinfeld's presentation of U-v(Lg) in the double Hall algebra setting, based on Schiffmann's work. We explicitly find out a collection of generators of the double composition algebra DC(Coh(X)) and verify that they satisfy all the Drinfeld relations. (C) 2012 Elsevier Inc. All rights reserved.