论文标题
与加权投影线相关的量子群集代数
Quantum cluster algebras associated to weighted projective lines
论文作者
论文摘要
令$ \ mathbb {x} _ {\ boldsymbol {p}，\boldsymbolλ} $为加权投影线。我们定义$ \ mathbb {x} _ {\ boldsymbol {p}，\boldsymbolλ} $的量子群集代数，并实现其专业版本，作为$ \ \ \ mthbb {x} _ {x _ Quantsmbol {\ boldsymbol的$ \ m mathbb {p} pronum的特殊版本。受\ cite {chen2021}的启发，我们证明了量子群集字符之间的模拟群集乘法公式。作为应用程序，我们获得了$ \ Mathbb {x} _ {\ boldsymbol {p}，\boldsymbolλ} $的格拉曼尼亚品种的多项式特性。最后，我们构造了几个bar-invariant $ \ mathbb {z} [ν^{\ pm}] $ - 基于投影线$ \ mathbb {p}^1 $的量子群集代数的基础，并显示了它与Kronecker Quiver的量子群集algebra的重合。
Let $\mathbb{X}_{\boldsymbol{p},\boldsymbolλ}$ be a weighted projective line. We define the quantum cluster algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbolλ}$ and realize its specialized version as the subquotient of the Hall algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbolλ}$ via the quantum cluster character map. Inspired by \cite{Chen2021}, we prove an analogue cluster multiplication formula between quantum cluster characters. As an application, we obtain the polynomial property of the cardinalities of Grassmannian varieties of exceptional coherent sheaves on $\mathbb{X}_{\boldsymbol{p},\boldsymbolλ}$ . In the end, we construct several bar-invariant $\mathbb{Z}[ν^{\pm}]$-bases for the quantum cluster algebra of the projective line $\mathbb{P}^1$ and show how it coincides with the quantum cluster algebra of the Kronecker quiver.
