A monoidal structure on the category of relative Hopf modules. (Q2885387)

Let \(B\) be a bialgebra in a braided monoidal category \(\mathcal C\) and let \(A\) be a left \(B\)-comodule algebra. The objects of the category \(^B\mathcal C_A\) of relative \((B,A)\)-Hopf modules have a left \(B\)-coaction and a right \(A\)-action that satisfy the appropriate variant of the Hopf module compatibility condition.NEWLINENEWLINE If \(A\) is also a coalgebra and there is a given morphism \(\varphi\) from \(B\otimes A\) to \(A\), it is possible to use these structure elements to define a potential right \(A\)-action on the tensor product of two relative \((B,A)\)-Hopf modules. This tensor product has in addition the standard left \(B\)-coaction.NEWLINENEWLINE The main result of the paper under review says that the category \(^B\mathcal C_A\) of relative \((B,A)\)-Hopf modules becomes a monoidal category in this way if, and only if, the coproduct and the additional morphism \(\varphi\) turn \(A\) into a braided bialgebra, i.e., a bialgebra in the category of left Yetter-Drinfeld modules over \(B\).