Irreducible Representation Of The Ore Extension Of U(g₂)

Hopf algebra is one of important algebra research fields.Ore extension is a kind of important ring extensions,which is widely used in the study of noncommutative rings.In recent years,this idea has been gradually applied to the study of Hopf algebras.With the integration of Hopf algebras and representation theory,many new construction methods have appeared.And many new Hopf algebras were constructed by using the Ore extension of Hopf algebra,and they are infinite dimensional Hopf algebras.It is an interesting and important topic to study the representations of these Hopf algebras,especially the irreducible representations of the Ore extensions of Hopf algebrasIn this paper,we study the irreducible representations of a class of Ore extensions of the enveloping algebra U(g2)of 2-dimensional non-abelian Lie algebra g2 over an algebraically closed field k of characteristic zero.It is well-known that there is a basis a,b in g2 such that[a,b]=a.There are two classes of generalized Hopf-Ore extensions of U(g2),and one of them is H=U(g2)(??,0,0,?),where a is a sclale,?? is a linear character of U(g2)determined by ?,and ? is a skew derivation of U(g2)with ?(g2)(?)g2.We organize the paper as follows.In Chapter one,we introduce the concepts of algebras,Lie algebras and their enveloping algebras,modules and module homomorphisms,and so on.We also introduce the structures of generalized Hopf-Ore extensions,the Hopf algebra structure of U(g2)and the Hopf-Ore extension H=U(g2)(??,0,0,?).In Chapter two,we study the finite dimensional irreducible representations of H=U(g2)(??,0,0,?).In Section one,we study the finite dimensional simple U(g2)-modules.It is shown that all finite dimensional simple U(g2)-modules are one dimensional.The structures of these modules are described and classified.In Section two,based on the structure of H,we study the finite dimensional simple H-modules for three cases.Firstly,we consider the case:?(a)=0,?(b)=?a,??k.In this case,when a?0,it is shown that every finite dimensional simple H-modules is one dimensional,and determined by a scale ? ? k.Moreover,the structures and classification of finite dimensional simple H-modules are given.When a=0,it is shown that every finite dimensional simple H-module is also one dimensional,and determined by an ordered pair of scales ?,??k.Meanwhile,we describe the structures of finite dimensional simple H-modules,and classify them.Then we consider the case:?=-1,?(a)=?b,?(b)=0,0???k.In this case,it is shown that for each positive integer n,there is exactly one n-dimensional simple H-module.The structure of such an n-dimensional simple H-module is described.Consequently,the finite dimensional simple H-modules are classified.Finally,we consider the case:?=-1,?(a)=a,?(b)=-b.In this case,it is proved that each finite dimensional simple H-module is one-dimensional,and exactly determined by a scale A E k.Then we describe the structures of the simple H-modules,and classify them.