Introduction

In the article we investigate balanced categories and balanced Hopf algebras. The close relation of balanced categories, balanced Hopf algebras and ribbon braids allows the use of diagrammatic morphisms in algebraic calculations for balanced Hopf algebras and categories and to discuss algebraic applications in knot theory.

In the first part we consider balanced categories and balanced Hopf algebras as well as ribbon and sovereign categories and Hopf algebras. Sovereign categories have been introduced in [9], sovereign Hopf algebras have been studied in [2]. From the reconstruction theoretical point of view they are the natural objects in relation with sovereign categories [2].

Strong sovereignity will be introduced and it will be shown that a Hopf algebra is strong sovereign if and only if it is a ribbon Hopf algebra. This result immediately implies the redundancy of the relations S(θ) = θ and θ2 = u S(u) for the twist element of a ribbon Hopf algebra (H,R, θ).

For every quasitriangular bialgebra a corresponding balanced bialgebra will be constructed by which we easily find an example of a balanced category related to a category of modules. Another example of a balanced category is the balanced construction out of a given monoidal category. Braided balanced categories with duality and braided sovereign categories are equivalent notations [6, 33, 25]. We provide an elementary proof of this fact using results on balanced categories with duality.