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

The quantum loop algebra Uv(Lg)Uv(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 gg. 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 Uv(Lg)Uv(Lg), for some gg with a star-shaped Dynkin diagram. In this paper we study Drinfeld’s presentation of Uv(Lg)Uv(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.