The Multiplication Formulas Of Quantum Cluster Algebras

This thesis aims to provide multiplication formulas of quantum cluster characters on quantum cluster algebras.There are four major parts in the thesis:(1)On abelian cate-gories with Ext -symmetry,we introduce weighted quantum cluster characters and prove corresponding multiplication formulas;(2)On 2-Calabi-Yau triangulated categories with cluster tilting objects,we introduce weighted quantum cluster characters and prove the corresponding multiplication formulas;(3)On subcategories of nilpotent module cate-gories of preprojective algebras,we study the relation between two multiplication for-mulas;(4)On cluster categories from hereditary algebras,we simplify the multiplication formulas.For abelian categories with Ext -symmetry,we first introduce chains of morphisms,chains of monomorphisms and types.For a short exact sequence in category,we prove that given chains of monomorphisms of lateral terms,appropriate chains of monomor-phisms of middle term exactly one-to-one correspond to a vector space.Then we intro-duce quantum cluster character of an object which is a linear map on certain vector space.Based on these,we introduce weight functions and weighted quantum cluster characters.For a given short exact sequence,we analyze relation between weighted quantum cluster characters of middle term and lateral terms.We prove given appropriate weight functions,such relation is an identity which is required multiplication formula.For 2-Calabi-Yau triangulated categories with cluster tilting object,we first consider triangle induced by a morphism and exact sequence lifted by the triangle through a functor from this category to a module category.For the exact sequence,we check with certain condition,given submodules of lateral terms,appropriate submodules of middle term ex-actly one-to-one correpond to a vector space.Then we introduce skew polynomial algebra and assign a skew polynomial for each object in category,called quantum cluster char-acter.Based on the character,we define weight functions and weighted quantum cluster characters.We calculate relation between weighted quantum cluster characters of mid-dle term and lateral terms.We prove given certain weight functions,such relation is an identity which is required multiplication formula.Notice that both multiplication formulas admit similar forms,but applicable cat-egories are different.We consider nilpotent module category of preprojective algebra and choose an appropriate subcategory.Such subcategory is abelian category with Ext -symmetry,and stabilization of the subcategory is 2-Calabi-Yau triangulated category with cluster tilting object.So in this subcategory,we define corresponding skew polynomial of first multiplication formula and compare it with the second one.Through choosing appropriate weight functions,we prove that both multiplication formulas coincide.Finally,we consider cluster category from heraditary algebra which is a 2-Calabi-Yau triangulated category with cluster tilting object.In this category,we simplify the multiplication formula step by step.If dimension of extension group equals to one,there is a proved multiplication formula.We compare the simplified multiplication formula and the existing one,and prove that they coincide.