Autonomous And Banded Unique Hom-Hopf Algebra

In this paper,we study the sovereign structure on finite dimensional monoidal HomHopf algebras,and give the definition of(∞)sovereign monoidal Hom-Hopf algebra,we also give a necessary and sufficient condition for a finite dimensional quasitriangular monoidal Hom-Hopf algebra(H,α,R)with bijective antipode to admit a ribbon structure.The paper has been divided as five sections.The first and the second sections are introduction and preliminary respectively.In the third section,we first introduce the notion of a sovereign monoidal Hom-Hopf algebra(H,α),and then consider the representation category Rep(H) over it,and proof:(H,α)is a sovereign monoidal Hom-Hopf algebra if and only if Rep(H)is a sovereign category.The fourth section is the dual case of third section,we introduce the notion of a cosovereign monoidal Hom-Hopf algebra and proof:(H,α)is a cosovereign monoidal Hom-Hopf algebra if and only if Corep(H)is a sovereign category.In the fifth section,we give a necessary and sufficient condition for a finite dimensional quasitriangular monoidal Hom-Hopf algebra(H,α,R)with bijective antipode to admit a ribbon structure.