On the structure of graded .-Hopf algebras

Let G be an abelian group, B the G-graded .-Hopf algebra with . being a bicharacter on G. By introducing some new twisted algebras (coalgebras), we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded .-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded .-Hopf algebra is invertible.