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Quantum cluster algebras associated to weighted projective lines (2207.02837v2)

Published 6 Jul 2022 in math.RT

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.

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