论文标题
类型$ \ widetilde {a} _ {2n-1,1} $的量子群集代数的原子基础
Atomic basis of quantum cluster algebra of type $\widetilde{A}_{2n-1,1}$
论文作者
论文摘要
令$ q $是类型$ \ widetilde {a} _ {2n-1,1} $的仿射手，而$ \ Mathcal {a} _ {q}（q）$是与估算Quiver $相关的量子群集代数（q，（q，q，（q，2,2，\ dots，2，2，2））$。我们证明了一些集群乘法公式，并推断出与$ q $的顶点相关的群集变量满足恒定系数线性关系的量子类似物。 We then construct two bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases $\mathcal{B}$ and $\mathcal{S}$ of $\mathcal{A}_{q}(Q)$ consisting of positive elements, and prove that $ \ Mathcal {B} $是原子基础。
Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the cluster variables associated with vertices of $Q$ satisfy a quantum analogue of the constant coefficient linear relations. We then construct two bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases $\mathcal{B}$ and $\mathcal{S}$ of $\mathcal{A}_{q}(Q)$ consisting of positive elements, and prove that $\mathcal{B}$ is an atomic basis.
