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

Rujing Dou, Yong Jiang, Jie Xiao

Number 50
Author Yong Jiang
Project C01
Year 2010

The quantum loop algebra U_{v}(\mathcal{L}\mathfrak{g}) was defined as a generalization of the Drinfeld's new realization of quantum affine algebra to the loop algebra of any Kac-Moody algebra \mathfrak{g}. Schiffmann \cite{S} has proved (and conjectured) that the Hall algebra of the category of coherent sheaves over weighted projective lines provides a realization of U_{v}(\mathcal{L}\mathfrak{g}) for those \mathfrak{g} associated to a star-shaped Dynkin diagram. In this paper we explicitly find out the elements in the Hall algebra \mathbf{H}(\Coh(\mathbb{X})) satisfying part of Drinfeld's relations, as addition to Schiffmann's work. Further we verify all Drinfeld's relations in the double Hall algebra \dh(\Coh(\mathbb{X})). As a corollary, we deduce that the double composition algebra is isomorphic to the whole quantum loop algebra when \mathfrak{g} is of finite or affine type.

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