无环主系数簇代数的范畴化
A categorification of acyclic principal coefficient cluster algebras
AI总结：
本文为无环主系数簇代数构造Frobenius范畴化，并通过部分稳定化定义extriangulated范畴实现范畴化，同时给出Ginzburg dg-代数的Frobenius版本双模3-Calabi-Yau性质。
摘要：
在先前的工作中，作者介绍了一种从内部Calabi-Yau代数的数据出发，为带有冻结变量的簇代数构造Frobenius范畴化的方法，该内部Calabi-Yau代数成为所得范畴中簇倾斜对象的内态代数。在本文中，我们为带有极化主系数（与主系数簇代数不同，它们通过添加更多冻结变量而有所区别）的簇代数构造合适的内部Calabi-Yau代数，并在无环情形下获得Frobenius范畴化。通过部分稳定化，我们进而定义extriangulated范畴（在Nakaoka和Palu的意义下），对无环主系数簇代数进行范畴化，而对于这些簇代数，Frobenius范畴化通常不存在。用于获得这些范畴化的许多中间结果在无环性假设不成立时仍然有效，我们将指出这一点，并且这些结果本身也很有趣。最值得注意的是，我们提供了Van den Bergh结果的Frobenius版本，即带势的箭图的Ginzburg dg-代数是双模3-Calabi-Yau的。
英文摘要：
In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi-Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi-Yau algebras for cluster algebras with polarised principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables), and obtain Frobenius categorifications in the acyclic case. Via partial stabilisation, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh's result that the Ginzburg dg-algebra of a quiver with potential is bimodule 3-Calabi-Yau.
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。
