Yetter–Drinfeld categories associated with a weak braided Hopf algebra

Abstract

In a previous paper, the authors introduced the monoidal category of left–left Yetter–Drinfeld modules over a weak braided Hopf algebra in a strict monoidal category. The main goal of this work is to define the categories of right–right, left–right and right–left Yetter–Drinfeld modules over a weak braided Hopf algebra and prove that there exists a categorical equivalence between all of them. We also establish the categorical equivalences by changing the weak braided Hopf algebra D by its (co)opposite. Finally, the general results are illustrated with an example coming from the projections of weak braided Hopf algebras.