Bott–Samelson Varieties and Poisson Ore Extensions
Bott–Samelson Varieties and Poisson Ore Extensions
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Bott–Samelson 品种和泊松矿石扩展
DOI：
10.1093/imrn/rnz127
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发表时间：
2016
影响因子：
1
通讯作者：
Jiang
中科院分区：
文献类型：
--
作者：
Bal'azs Elek;Jiang
我们证明了对于复半简单李群$G$的任意$n维bot - samelson变体，在多项式代数$ a ={\mathbb{C}}[z_1, \ldots, z_n]$上有$2^n$泊松括号，每一个都是迭代泊松Ore扩展，其中一个是Goodearl-Yakimov意义上的对称泊松Cauchon-Goodearl-Letzter （CGL）扩展。我们用根弦和李代数的结构常数来表示泊松括号。由此可见，所有广义Bruhat单元的坐标环都有对称泊松CGL扩展的表示。本文建立了广义Bruhat细胞的基础，并为其在可积系统、聚类代数、全正性和泊松变异的环退化等方面的应用奠定了基础，其中一些在引言部分进行了讨论。
We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.