Quantum cluster algebras associated to weighted projective lines

Abstract

Let ${\mathbb{X}}_{\mathbf{p},\mathbf{\lambda }}$ be a weighted projective line. We define the quantum cluster algebra of ${\mathbb{X}}_{\mathbf{p},\mathbf{\lambda }}$ and realize its specialized version as the subquotient of the Hall algebra of ${\mathbb{X}}_{\mathbf{p},\mathbf{\lambda }}$ 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 ${\mathbb{X}}_{\mathbf{p},\mathbf{\lambda }}$ . In the end, we construct several bar-invariant $\mathbb{Z}[{\nu }^{\pm }]$-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.

38 pages