Applications Of Cluster Algebra In Quantum Affine Algebra And Poisson Geometry

Cluster algebra has close connections to many areas including quantum groups,poisson geometry,and integrable systems,especially,cluster algebra can be ap?plied to study the canonical basis of quantum groups and the compatible of cluster structure and poisson structure can be applied to study the cluster structure of Lie groups.They are four chapters in this dissertation.In chapter 1,we introduce the research background and preliminary knowledge.In chapter 2,we introduce two systems of equations called M-systems(dual M-systems)of types An and Bn respectively,and make a connection between M-systems(dual M-systems)and cluster algebras.We also prove that the Hernandez-Leclerc conjecture is true for minimal affinizations of types An and Bn.In chapter 3,by using the theory that cluster structure is compatible with poisson structure,we find the cluster structure of Lie group*SO(3,C),Drinfeld double of SO(3,C)and SO(5,C).As the applications,they confirm that conjecture proposed by Gekhtman,Shapiro and Vainshtein is true for simple complex Lie group SO(3,C),Drinfeld double of SO(3,C),SO(5,C).In chapter 4,by introducing Schur Positivity for minimal affinizations of types An,Bn,G2,we prove that the conjecture that proposed by Chari,Fourier and Sagaki is true for minimal affinizations of types An,Bn,G2.