Geometric Representation Theory Seminar | Quantum cluster algebras associated to weighted projective lines

Abstract：

In the first part of this talk, I will briefly introduce categorification of acyclic quantum cluster algebras by cluster categories of acyclic quivers, based on the work of Fan Qin. In the second part, I will explain how to categorify certain quantum cluster algebras using cluster categories of coherent sheaves on weighted projective lines. Concretely, we firstly define specialized quantum cluster characters of objects in the cluster category over finite fields and then show a cluster multiplication formula, which gives rise to mutation relations of quantum cluster algebras. Moreover, we can show quantum cluster characters of indecomposable rigid objects are generic and then coincide with quantum cluster variables. If time permitted, I will also introduce some applications of this categorification, such as finding good bases.