标题： T-对数-辛对数规范泊松结构的形变及对称泊松CGL扩张 显示英文标题

标题： Deformations of T-log-symplectic log-canonical Poisson structures and symmetric Poisson CGL extensions

摘要： 对于复代数环面$\mathbb{T}$，我们研究$\mathbb{T}$-不变的泊松变形，即在$\mathbb{C}^n$上的$\mathbb{T}$-对数辛对数规范泊松结构。我们证明，在适度假设下，每个没有$(\mathbb{C}^\times)^n$-不变分量的$\mathbb{T}$-不变的一阶变形都是无阻碍的。 作为应用，我们证明了 $\mathbb{T}$-对数辛-对数规范的泊松结构在 $\mathbb{C}^n$上的一个特殊类，即由所谓的 $\mathbb{T}$-作用数据定义的结构，可以被规范地变形为 $\mathbb{T}$-不变的代数泊松结构在 $\mathbb{C}^n$上，这些结构是（强）对称的CGL扩展（在Goodearl-Yakimov的意义下为 $\mathbb{C})$）。特别是，我们从任何与可对称化广义Cartan矩阵 $A$相关的简单根序列构造（强）对称的CGL扩展。 当$A$为有限类型时，我们的构造恢复了Bott-Samelson单元和广义Schubert单元上的标准泊松结构，这些结构与这些单元上的(标准)簇代数结构密切相关。 显示英文摘要

摘要： For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of $\mathbb{T}$-log-symplectic log-canonical Poisson structures on $\mathbb{C}^n$. We show that, under mild assumptions, every $\mathbb{T}$-invariant first-order deformation with no $(\mathbb{C}^\times)^n$-invariant component is unobstructed. As an application, we prove that a special class of $\mathbb{T}$-log-symplectic log-canonical Poisson structures on $\mathbb{C}^n$, namely those defined by the so-called $\mathbb{T}$-action data, can be canonically deformed to $\mathbb{T}$-invariant algebraic Poisson structures on $\mathbb{C}^n$ that are (strongly) symmetric Poisson CGL extensions (of $\mathbb{C})$ in the sense of Goodearl-Yakimov. In particular, we construct (strongly) symmetric CGL extensions from any sequence of simple roots associated to any symmetrizable generalized Cartan matrix $A$. When $A$ is of finite type, our construction recovers the standard Poisson structures on Bott-Samelson cells and on generalized Schubert cells, which are closely related to the (standard) cluster algebra structures on such cells.