Cluster algebra structures on Poisson nilpotent algebras
Cluster algebra structures on Poisson nilpotent algebras
复制标题
泊松幂零代数上的簇代数结构
DOI：
10.1090/memo/1445
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发表时间：
2018
影响因子：
1.9
通讯作者：
M. Yakimov
中科院分区：
文献类型：
--
作者：
K. Goodearl;M. Yakimov
在泊松李群理论和泊松齐次空间中出现的各种变异坐标环都属于大的、公理定义的对称泊松幂零代数类，如对称Kac-Moody群的Schubert胞的坐标环、bot - samelson变异的仿射图、双Bruhat胞的坐标环（最后一种情况是在局部化之后）。证明了每一个在一定标量上满足温和条件的对称泊松幂零代数都是与相应的上聚类代数相一致的聚类代数的正则同构，而不需要额外的冻结变量的局部化。构建的簇结构与Gekhtman、Shapiro和Vainshtein意义上的泊松结构是相容的。证明了所有泊松幂零代数都是等变泊松唯一分解域。它们的种子由泊松UFDs链的泊松素元素序列构造；突变矩阵是根据潜在泊松结构从线性系统有效确定的。建立了泊松素数序列的唯一性、存在性、突变性等性质。
Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.