A Restrict PBW Basis In Hopf Quiver Algebra

This thesis is related to the theory of the basis of Hopf quiver algebra. Ourfocus is the multiplication matrix of the path coalgebra, which is constructed by themultiplication of paths. We show in this paper how to express the PBW basis of somealgebras in terms of path, compute the dimension of3-order connect Nichols algebra onfnite cyclic groups in three cases and give the specifc representation of correspondingPBW basis, which is of importance.The thesis is divided into three sections.In the frst section, we begin by some basic defnition and properties of quiverthat are useful for this paper. And then we introduce the most important defnition,path multiplication matrix, by considering the multiplication rule of path in pathcoalgebra.The path multiplication greatly reduce the mistake which is taken by themultiplication between the path and path.And we will give two examples which canconstruct the convenience of the constructed path multiplication matrix,which cangreatly help some diferent the path multiplication.In the second section, we focus on how to represent PBW basis in terms of pathsand some important theorem and conclusions are given. Some of those are as follows:（i）We give the formula of （aderi）ejby means of paths;and we also proof this is a hard supper word.（ii）We proof that [x₁x₂,···, x₂] is also a super hard word.And we also give the path’sIn the end, we obtain the dimensions of connected Nichols algebras with rank3over fnite cycle groups.And give some general expressions.