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Bridgeland的Dynkin类型Hall代数的sheaf实现

Sheaf realization of Bridgeland's Hall algebra of Dynkin type

School of Mathematical Sciences, Peking University（北京大学数学科学学院）；The University of Hong Kong（香港大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Max Planck Institute for mathematics（马克斯·普朗克数学研究所）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）

Jiepeng Fang, Yixin Lan, Jie Xiao

备注：
In the new version, we add a new result (Theorem 8.1): the $\mathbb{Z}[v,v^{-1}]$-algebra $\mathcal{D}\mathcal{K}^{ss,*,\mathrm{red}}$ constructed by semisimple complexes and the restriction functor in section 6, is isomorphic to the extension counting integral form of the generic Bridgeland's Hall algebra and is isomorphic to the Poisson integral form of the quantum group
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AI总结：

本文基于Bridgeland的Dynkin类型Hall代数，通过sheaf方法实现了量子群的范畴化，并获得了具有正性的基和整数形式的范畴化。

摘要：

作为[6]中的一个结果，Bridgeland通过有限域上quiver表示的双周期投射复形的Ringel-Hall代数的局部化，实现了量子群$\mathbf{U}_v$。在本文中，我们将Lusztig对nilpotent部分$\mathbf{U}_v^+$的范畴构造推广到Bridgeland的Dynkin类型Hall代数。特别是，我们得到了双周期投射复形的Ringel-Hall代数的一个基，该基具有正性；我们范畴化了通用Bridgeland Hall代数的整数形式，该形式与$\mathbf{U}_v$的Poisson积分形式同构，并获得了该整数形式的$\mathbb{Z}[v,v^{-1}]$-基。

英文摘要：

As one of results in [6], Bridgeland realized the quantum group $\mathbf{U}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part $\mathbf{U}_v^+$ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of $\mathbf{U}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form.

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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