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

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

2010-02-05

arxiv.org/abs/1002.1316v1

Mathematics

Representation Theory

31pages

Scientific paper

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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