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

The quantum loop algebra 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 g. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is c...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2012-12, Vol.231 (5), p.2593-2625
Main Authors: Dou, Rujing, Jiang, Yong, Xiao, Jie
Format: Article
Language:English
Subjects:
Coherent sheaf
Drinfeld’s presentation
Hall algebra
Quantum loop algebra
Weighted projective line
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Summary:The quantum loop algebra 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 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 Uv(Lg), for some g with a star-shaped Dynkin diagram. In this paper we study Drinfeld’s presentation of 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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2012.07.026