We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the formTB(M)/I, whereTB(M) is a path algebra with a graded Hopf algebra structure, andIis a graded admissible Hopf ideal.