Rank Of An Infinite-dimensional Pointed Hopf Algebra,

Let H be a Hopf algebra over the field k,H₀(?)H₁(?)H₂(?)···be the coradical filtration of H.Assume that H₀ is a subHopf algebra of H.When H is generated by H₁ as an Algebra,Krop and Radford assigned a measure of complexity to H which is called the rank of H in[13].They obtained a presentation of finite dimensional pointed Hopf algebras of rank one over k by generators and relations,where k is an algebraically closed field with characteristic 0.In this paper,two kinds of pointed Hopf algebras are studied which are both infinite dimensional,and the relation between them is also given.Firstly,we discuss the rank of the Hopf Ore extension for the group algebra.Next,we classify the infinite dimensional pointed Hopf algebra of rank one.In section 1,we recall some basic concepts and results about Hopf algebra,the rank of Hopf algebra,Hopf Ore extension and so on,which are used later in the paper.In section 2,let kG(x,a,δ) be the Hopf Ore extension for the group algebra kG.We discuss the rank of H and prove that:when x(a) is a primitive n-th root of unity(n≥2), H₁= H₀ + H₀x + H₀xⁿ,that is,the rank of H is 2;otherwise,H₁ =H₀ ? H₀x,i.e.,the rank of H is1.In section 3,let H be an infinite dimensional pointed Hopf algebra of rank one,G=G(H) be the set of group-like elements in H.We prove that there exist some a∈G,x∈H\H₀ such that△(x) = x(?)a+1(?)x.Hence H₁=H₀?H₀x.We also show that H is exactly the Hopf Ore extension of the coradical kG.According to the parameters,H is classified into three types in section 3.In section 4,we study the representation of H which belongs to the third type.If G is abelian,we show that: when |x| 1=∞,every finite dimensional simple H-module must be a weight module and 1 dimensional;when |x|= n＜∞,it must be a weight module,and 1 or n dimensional. Moreover,these simple modules are determined.Finally,we define the Verma modules over H and discuss the properties of these modules.These Verma modules are indecomposable weight modules.In particular,they are the projective objects in the weight module category W.