Classification Of Hopf Algebras Of A Given Dimension

In 1975,Kaplansky put forward ten conjectures about Hopf algebra.Affected by these conjectures,mathematicians have had great passion for the classification of Hopf algebras of a given dimension over a field since the 1990s and they obtained a series of results.This paper is a survey about this problem.In this paper,we summarize the classification results for semisimple Hopf algebra and pointed Hopf algebra and we follow the proof of one of these results.In chapter 1,we briefly introduce the origin and development of Hopf algebra,Kaplansky’s ten conjectures and the background of the classification problem.We also introduce three types of Hopf algebras that mathematicians focus on:semisimple Hopf algebra,pointed Hopf algebra and Hopf algebra with Chevalley property.In chapter 2,we give the basic concepts and conclusions about Hopf algebra.In particular,we give the definitions of the three types of Hopf algebras introduced in the first chapter.In chapter 3,we systematically summarize the main results obtained for the classification problem:on the one hand,we divide chapter 3 into different sections according to the dimension;on the other hand,in each section,we first describe the development history of the given dimension,then we summarize the theorems.In the first seven sections,we describe the classification of n-dimensional Hopf algebra in detail,where n has at most three prime factors.Good progress has been made in the classification of these dimensions.In fact,the classification of Hopf algebra of dimension p,p~2 and 2p~2 has been completely solved.In the eighth section,we briefly describe the results about other dimensions.Because these results are relatively fragmented,we gather them in one section.At the end of this chapter,we briefly introduce the classification over a field of characteristic p.Siu-Hung Ng proved that a non-semisimple Hopf algebra of dimension p~2 Aust be a Taft algebra.We elaborate on his proof in chapter 4 to reveal how to classify Hopf algebras by studying antipode.