The Accompanying Representation Of The Taft Algebra And Its Killing Type

In early 40's,a German mathematician,H.Hopf in the study of topological properties of Lie group constructed algebra system with an algebra structure and co-algebra,which is now called Hopf algebra.Kaplansky in 1975 summarized the latest research results of the Hopf algebra and proposed ten conjectures,which strongly impet to the development of Hopf Algebra in recent years,in particular,to the quantum group(a special class of Hopf algebra)with the deep connection between quantum groups and Yang-Baxter equation in quantum mechanics.Now Hopf algebra has become a branch of algebra closely associated with the disciplines such as mathematics and physics.While the Taft Hopf algebra as an important class of non semisimple,non-commutative and noncommutative Hopf algebra,offers us a better theoretical framework and research ideas in the Hopf algebra theory.This thesis mainly studies the adjoint representations of the Taft Hopf algebras and its decomposition,the Killing form and Killing radical of the Taft Hopf algebras.Using the decomposition we completely give the classification of ideals of Taft Hopf algebra.The results show that every ideal is a principal ideal,while the Killing radical of Taft Hopf algebras is the same as Jacobson radical.This thesis is divided into three chapters.The first chapter reviews the basic concepts about Hopf algebra,adjoint action,Taft Hopf algebra and so on,which will be used in this paper.The second chapter studies the decomposition Taft Hopf algebra in the adjoint action.Using thisdeccomposition we obtain the classification of the ideals of Taft Hopf algebra and every ideal is a principal ideal.In the third chapter,we first review the definition of Killing form of Hopf algebra and its basic properties;secondly the explicit calculation Killing form of Taft Hopf algebra;finally,we prove that the radical of Killing form is the same as the Jacobson radical.