Two Kinds Of Classifications Of Hopf Algebras And Relative Topics

Hopf algebra is an important family of algebras,which is closely related to quantum groups,representation theory,noncommutative geometry,mathematical physics,and so on.It is of interest in Hopf algebras theory to investigate Hopf algebra structures on certain algebras.In ring theory,Ore extension and Dorroh extension are two important ring extensions,which are widely used in the study of these structures and properties of various rings.Now these methods are widely used in the research of Hopf algebras and quantum groups.In particular,we can consider the structures of Hopf algebras on Ore extensions and Dorroh extensions.In this Doctoral thesis,we will focus on the Ore extensions and Dorroh extensions of Hopf algebras,and study their structures,properties,representations,and so on.The thesis is arranged as follows.In Chapter 1,we introduce some notations and basic concepts,including Ore exten-sions,Hopf-Ore extensions,Dorroh extensions of algebras and coalgebras and so on.In Chapter 2,We first introduce the primitive cohomology of coalgebra and re-describes Taft-Wilson theorem.Then we study the Hopf algebra structures on an Ore extension H=A[z;?,?]of a Hopf algebra A such that ?(z)=z(?)r1+r2(?)z+v(z(?)z)+w,where r1,r2 ? A,v,w ? A(?)A.For various cases of v,the necessary and sufficient con-ditions for an Ore extension of a Hopf algebra to have some Hopf algebra structure are given.This generalizes the concept of Hopf-Ore extension.Finally,we study the struc-tures and properties of Hopf-Ore extensions on group algebras and Hopf-Ore extensions of the enveloping algebras of Lie algebras.In Chapter 3,based on Chapter 2,we consider Hopf-Ore extensions when v=0,w=x(?)y.Similarly to Chapter 2,we study such Hopf algebra structures on Ore extensions of Hopf algebras.For some Lie algebras g and sln,the Hopf-Ore exten-sions of their e enveloping algebras U(g)and U(sln)are classified,respectively.Final-ly,we discuss the finite dimensional irreducible representations of Hopf-Ore extension H=U(g2)(?1,a,b,?2)of the enveloping algebra U(g)of 2-dimensional noncommutative Lie algebra.In Chapter 4,we study Dorroh extensions of algebras(not necessarily having an identity)and Dorroh extensions of coalgebras(not necessarily having a counit).Their structures are described.Some properties of these extensions are presented.We also in-troduce the finite duals of algebras and modules.Using these finite duals,we determine the dual relations between the two kinds of Dorroh extensions.Combining of two kinds of Dorroh extensions,we also study Dorroh extensions of bialgebras and Hopf algebras.Let(H,I)be both a Dorroh pair of algebras and a Dorroh pair of coalgebras.We give necessary and sufficient conditions for H(?)I to be a bialgebra and a Hopf algebra,respec-tively.We also describe all ideals of Dorroh extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras,and compute these ideals and subcoalgebras for some concrete examples.