Hopf Quivers And Braided Hopf Algebras

It is well known that Hopf quivers admit graded Hopf algebra structures. In this paper, Hopf quivers and integrals of braided Hopf algebras are discussed.Firstly, it is proved that any Hopf quiver is the union of some basic Hopf quivers and any Hopf quiver of a finite cyclic group is the union of some directed cycles. Since a finite abelian group is the direct product of some finite cyclic groups, it is obtained that any Hopf quiver is the direct product of some directed cycles. Moreover, all simple basic Hopf quivers of non-abelian groups with order≤ 10 are provided.Secondly, graph properties of simple undirected Hopf quivers are discussed. Examples are given to show the following facts. There exists a Hopf quiver with odd(even) Hopf degree. There exists a regular graph with odd(even) degree not being a Hopf quiver. There exist a Hopf quiver being an Eulerian graph and one not. The relation between Hopf quivers and Hopf algebras is introduced.Finally, the relation between integrals and the maximal rational H^d-submodule H (drat) of H^d is found . The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category ^?yD are given.