Quiver grassmannians, quiver varieties and the preprojective algebra
Quiver grassmannians, quiver varieties and the preprojective algebra
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DOI：
10.2140/pjm.2011.251.393
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发表时间：
2009-09
影响因子：
0.6
通讯作者：
Alistair Savage;P. Tingley
中科院分区：
文献类型：
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作者：
Alistair Savage;P. Tingley
箭袋在代数表示论中发挥着重要作用，其关键成分是路径代数和预投影代数。箭袋草曼尼亚是路径或预投影代数的固定模块的子模块的变体。在本文中，我们详细研究了这些对象。我们证明，与某些单射模的子模相对应的箭袋格拉斯曼量与中岛的拉格朗日箭袋变体同胚，这些拉格朗日箭袋变体已在几何表示理论的背景下得到了很好的研究。然后，我们通过寻找与第一作者介绍的 Demazure 箭袋变种同构的箭袋草曼尼亚，以及与 Nakajima 定义的分级/循环箭袋变种同构的其他箭袋草变种来完善这一结果。 Demazure 箭袋grassmannians 允许我们在原投影代数的局部幂零模范畴中描述单射对象。最后，我们将我们的构造与使用射影代替单射的 Lusztig 的类似构造联系起来。在当前论文第一版发布后添加的附录中，我们展示了 Shipman 的后续结果如何暗示上述同构实际上是代数簇的同构。
Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives. In an appendix added after the first version of the current paper was released, we show how subsequent results of Shipman imply that the above homeomorphisms are in fact isomorphisms of algebraic varieties.