论文标题
集群代数的泊松几何和阿祖玛亚基因座
Poisson geometry and Azumaya loci of cluster algebras
论文作者
论文摘要
群集代数理论中有两种主要类型的对象类型：上群集代数$ {\ boldsymbol {\ MathSf u}} $，其gekhtman-shapiro-vainshtein poisson poisson poisson brackets及其unity量子的根及其根的根源$ {\ boldsymbol {\ boldsymbol {\ boldsymbol {\ nathssf i}在泊松一侧，我们证明（没有任何假设）每个有限生成的上层群集代数$ {\ boldsymbol {\ boldsymbol {\ mathsf u}} $，其GSV Poisson结构始终具有Zariski的Zariski开放式镜头，并具有同义词的叶子，并给出了它的明确描述。在量子方面，我们描述了量化的完全azumaya基因座$ {\ boldsymbol {\ MathSf u}} _ \ varepsilon $，假设$ {\ boldsymbol {\ boldsymbol {\ natersf a}}} u}} _ \ varepsilon $和$ {\ boldsymbol {\ mathsf u}} _ \ varepsilon $是有限生成的代数。所有结果允许冷冻变量倒置或不倒。
There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${\boldsymbol{\mathsf U}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${\boldsymbol{\mathsf U}}_\varepsilon$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra ${\boldsymbol{\mathsf U}}$ with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the quantizations ${\boldsymbol{\mathsf U}}_\varepsilon$ under the assumption that ${\boldsymbol{\mathsf A}}_\varepsilon = {\boldsymbol{\mathsf U}}_\varepsilon$ and ${\boldsymbol{\mathsf U}}_\varepsilon$ is a finitely generated algebra. All results allow frozen variables to be either inverted or not.
