Quantum cluster algebras associated to weighted projective lines

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.

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