| In recent 40 years,with the development of quantum groups,the deep relation between quantum group and Yang-Baxter equation in theoretical physics is established.Hopf algebras and quantum groups have developed into branches that are closely related to algebraic groups,Lie algebras,representation theory,mathematical physics,and quantum mechanics.As an important example of quantum groups,the Radford quantum group is a class of Hopf algebras which are non-commutative and non-cocommutative.The research on it has opened up new ideas for further study on the quantum groups and Hopf algebras.On the other hand,the adjoint action of Hopf algebra plays an important role in the study of Hopf algebra.For a given Hopf algebra,how to give the direct sum decomposition of the Hopf algebra under the adjoint action and determine the generators of ideal of the Hopf algebra,is an open topic in the theory of Hopf algebra.This thesis mainly studies all forms of indecomposable submodules of Radford quantum groups under the adjoint action,presents the decomposition of Radford quantum groups under the adjoint action,and then proves that any ideal of Radford quantum groups is generated by one element.The thesis is divided into three chapters.The first chapter reviews the relevant concepts and definitions about bialgebra,Hopf algebra,adjoint action,and so on,which will be used in this paper.In chapter two,we give all indecomposable submodules of Radford quantum groups under the adjoint action.In the third chapter,using the decomposition formula of Radford quantun group under the adjoint action,we prove that each ideal of Radford quantum group can be generated by one element.Finally application,we give all the ideals of the Sweedler 4-dimensional Hopf algebra. |
