Cluster algebras III: Upper bounds and double Bruhat cells
Cluster algebras III: Upper bounds and double Bruhat cells
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DOI：
10.1215/s0012-7094-04-12611-9
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发表时间：
2005-01-15
影响因子：
2.5
通讯作者：
Zelevinsky, A
中科院分区：
文献类型：
--
作者：
Berenstein, A;Fomin, S;Zelevinsky, A
我们开发了一种新的方法，集群代数的概念的基础上，上集群代数定义为一个交叉的洛朗多项式环。加强[7]中建立的Laurent现象，我们证明了在“非循环性”的假设下，一个簇代数与它的tipper对应物重合，并且是双生成的;在这种情况下，我们还描述了它的定义理想，并构造了一个标准单项式基.证明了半单复李群中任意双Bruhat胞元的坐标环自然同构于一个根据相关组合数据明确定义的tipper簇代数.
We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [7], we show that under an assumption of "acyclicity," a cluster algebra coincides with its tipper counterpart and is finitely generated; in this case, we also describe its defining ideal and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an tipper cluster algebra explicitly defined in terms of relevant combinatorial data.