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Drinfeld量子圈代数表示的Hall代数方法

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

Rujing Dou, Yong Jiang, Jie Xiao

摘要:

量子圈代数$U_{v}(\mathcal{L}\mathfrak{g})$是将量子仿射代数的Drinfeld新实现推广到任意Kac-Moody代数$\mathfrak{g}$的圈代数上. Schiffmann证明了加权射影线上相干层范畴的Hall代数与量子圈代数$U_{v}(\mathcal{L}\mathfrak{g})$密切相关,对于某些具有星形Dynkin图的$\mathfrak{g}$。在本文中,我们研究Drinfeld的介绍$U_{v}(\mathcal{L}\mathfrak{g})$在双霍尔代数设置,Schiffmann的工作的基础上。我们明确地找到了双复合代数$\mathbf{DC}(\Coh(\mathbb{X}))$的生成元的集合,并证明了它们满足所有的Drinfeld关系.

英文摘要:

The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ 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 $\mathfrak{g}$. 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 $U_{v}(\mathcal{L}\mathfrak{g})$, for some $\mathfrak{g}$ with a star-shaped Dynkin diagram. In this paper we study Drinfeld's presentation of $U_{v}(\mathcal{L}\mathfrak{g})$ in the double Hall algebra setting, based on Schiffmann's work. We explicitly find out a collection of generators of the double composition algebra $\mathbf{DC}(\Coh(\mathbb{X}))$ and verify that they satisfy all the Drinfeld relations.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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