论文标题
月桂素现象代数的某些基本特性
Some elementary properties of Laurent phenomenon algebras
论文作者
论文摘要
令$σ$为laurent现象（lp）等级$ n $，$ \ mathcal {a}（σ）$，$ \ mathcal {u}（σ）$和$ \ mathcal {l}（σ）$分别是其相应的lurent Algebra上限和下限。我们证明，$ \ Mathcal {a}（σ）$的每种种子都是由其群集唯一定义的，并且具有$ n-1 $ common common common common common commun cluster变量的$ \ Mathcal {a}（σ）$的任何两个种子都通过一个步骤相互连接。本文中的方法还适用于（完全符号对称）群集代数。此外，我们表明，当每个交换多项式与其Exchange Laurent laurent多项式（$σ$）重合时，$ \ Mathcal {u}（σ）$在种子突变下是不变的。此外，我们获得了$ \ mathcal {l}（σ）$的标准单基础。我们还证明，在某些条件下，$ \ MATHCAL {U}（σ）$与$ \ Mathcal {l}（σ）$重合。
Let $Σ$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(Σ)$, $\mathcal{U}(Σ)$ and $\mathcal{L}(Σ)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(Σ)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(Σ)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(Σ)$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $Σ$. Besides, we obtain the standard monomial bases of $\mathcal{L}(Σ)$. We also prove that $\mathcal{U}(Σ)$ coincides with $\mathcal{L}(Σ)$ under certain conditions.
