Quiver varieties and cluster algebras
Quiver varieties and cluster algebras
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DOI：
10.1215/0023608x-2010-021
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发表时间：
2009-05
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通讯作者：
H. Nakajima
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作者：
H. Nakajima
受埃尔南德斯和Leclerc最近的一个猜想的启发[arXiv：0903.1452]，我们将一个Fomin-Zelevinsky簇代数[arXiv：math/0104151]嵌入到对称Kac-Moody李代数g的量子圈代数U_q（Lg）的表示范畴的Grothendieck环R中，作者早先通过分次簇上的反常层[arXiv：math/9912158]研究了这一点.控制图像的分级的类簇可以与Lusztig用来定义规范基的簇相识别。簇单项式形成由R中的简单模类给出的基的子集，或Lusztig的对偶标准基。当有一个种子具有一个二分多项式时，聚类单项式的正性和线性独立性（可能还有许多其他）图[arXiv：math/0104151]作为结果。简单的模块对应于集群单项式分解成张量积的“素”简单的根据集群扩展。
Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.