Classification Of Ideals Of Finite-Dimensional Pointed Hopf Algebras Of Rank One

Hopf algebras are important branches of algebras,which originate from alge-braic topology and algebraic group theory.The classification of ideals of algebras,especially Hopf algebras,has always been concerns of mathematicians.Among many algebras,there is a special class of algebras called principal ideal rings(i.e.each ideal is generated by a single element),which has a concise and beautiful structure.In recent decades,mathematicians have proved that many classes of Hopf algebras are principal ideal rings.Especially,in recent years,mathematicians also raise an open question:under what conditions Hopf algebras are principal ideal rings.It seems that this problem is interesting,momentous but difficult to be solved in the near future.In this dissertation,we study the classification of ideals of finite-dimensional pointed Hopf algebras of rank one over an algebraically closed field of characteristic zero.In Chapter 1,we introduce some notations and basic concepts and review the origin and development histroy of Hopf algebras.In particular,we recall the development process of the finite-dimensional pointed Hopf algebras of rank one.And we look back on algebras which have been proven to be principal ideal rings in history.Then the main results and preliminary knowledge of this paper are briefly listed.In Chapter 2,we first review the structure and classification of indecomposable modules of H,where H is a finite-dimensional pointed Hopf algebra of rank one.We find that every indecomposable submodule of H is generated by a special element under the adjoint action.Hence we prove that H is a principal ideal ring and we describe the generators of all ideals of H.Then we use the primitive central orthog-onal idempotents of the group algebra to characterize the generators of annihilator ideals of all indecomposable H-modules.When H is of nilpotent type,we also give the generators of completely prime ideals of H.In particular,since Radford Hopf algebras are a special class of finite-dimensional pointed Hopf algebras of rank one,we give the general forms of generators of all ideals of Radford Hopf algebras and calculate the generators of annihilator ideals of indecomposable modules and the generators of completely prime ideals of Taft Hopf algebras.In addition,we also calculate the intersection of some annihilator ideals of indecomposable modules of Radford Hopf algebras.In Chapter 3,we give generators of all ideals of 9-dimensional,16-dimensional and 8-dimensional Radford Hopf algebras,and use the generators to classify the ideals of these three kinds of low-dimensional Radford Hopf algebras.In Chapter 4,we give classification of all ideals of pointed Hopf algebra of rank one of nilpotent type over Klein 4-group.And we show that there exists a finite-dimensional pointed Hopf algebra which is not a principal ideal ring.