Quantum cluster algebras associated to weighted projective lines

TL;DR

This paper introduces quantum cluster algebras associated with weighted projective lines, establishes their properties via Hall algebras, and constructs bases linking to classical quantum cluster algebras.

Contribution

It defines quantum cluster algebras for weighted projective lines, proves a cluster multiplication formula, and constructs bar-invariant bases connecting to known quantum cluster algebras.

Findings

Polynomial property of Grassmannian variety cardinalities

Realization of quantum cluster algebra as Hall algebra subquotient

Construction of bar-invariant bases for quantum cluster algebra of P^1

Abstract

Let $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ be a weighted projective line. We define the quantum cluster algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ and realize its specialized version as the subquotient of the Hall algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ 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 $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ . 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.